A time travel story can simply use time travel as a vehicle to get the hero to the Adventure Towns, or the phlebotinum involved can be a key plot driver. No matter what story type the hero is going to need a Time Machine or Time Master to get around. Time Travel stories seem to fall into several categories:
Characters go to the past! In the past, they change history: If they do so by accident, it well may end the story with a Karmic Twist Ending; alternately, it will set the real plot in motion by requiring the characters to Set Right What Once Went Wrong.
On the other hand, they may have set out to change history intentionally, so that the events that create their future/present — and, thus, the conditions that prompted them to go back in time — never happened, basically the same set up as above, but without the initial "accident."
Characters go to the future! Upon returning to the past, they are able to fight fate and prevent the events of the future (seeing which prompted them to try to prevent the events of the future in the first place) from occurring.
Reset Button: The characters go through a world of crap, or somebody "changes history", and they resort to time travel to fix it. If they succeed, the time-line fixes itself and the characters awaken having no knowledge that anything was ever different. Occasionally, only the time-travellers remember — at least, the ones who were alive at the point of fix. If they don't succeed, the series has just received a Re Tool or Story Reset.
Trapped in the Past: The characters are stuck in another time with no way of return (a.k.a. forced to take The Slow Path) and must choose between quietly living out their lives without changing history or working to change the world to their (and the natives') benefit. You'd be amazed how few people seem to pick the first option.
Alternate Timeline: The characters time-travel has split their universe in twain. There's the universe they're in (that they've "changed") and the universe they're not in. (the "old" universe that wasn't changed.)
In Simoun, time travel requires the successful completion of the Emerald Ri Maajon, an extremely dangerous maneuver that can only be accomplished by a pair of the most skillful pilots with a powerful emotional bond with each other. Failed attempts are generally fatal, with explosive consequences.
In Universal War One, scientists build a space station that accidentally opens a wormhole, allowing limited time travel. Then Kalish solves the equations that allow anybody to travel through time and space without limitation.
The Billy & Mandy story "Better Luck Next Time" (Cartoon Network Block Party #45) has Billy messing with Grim's demonic cuckoo clock. It sends him and Mandy through time where they meet a mysterious cloaked figure that tells them to return to where they started. It backfires thanks to Billy's blundering. The cloaked figure turns out to be Grim in the future, resigned to being consigned to the two forever.
In Donnie Darko people in the future will be able to mess with the past without leaving the future via machine. Such meddling causes alternate universes which need destroyed or they'll erase the future-people's universe.
In Back to the Future, you needed a way to generate 1.21 gigawatts of power, such as nuclear fuel or a lightning strike, and a ground speed of 88 miles per hour.
In the Terminator series, only organic things could be sent through time. No weapons or clothes or anything but the time traveller.
Rather conveniently forgotten by T2. The only thing that allowed the original T-800 model 101 back through time was the fact it was a shell surrounded by living tissue. It was, to quote Kyle Reese, "something about the field generated by a living organism'. Nothing dead will go." Therefore how exactly the T-1000 which was liquid metal managed to travel through is just, well, kinda unexplained.
In Jean Claude Van Damme's Timecop, there's a federal agency responsible for going after people who attempt to go back in time. He winds up having to go back in time himself to save his wife from dying, which is what he was hired to keep other people from doing.
Star Trek 4, 7, 8 and 11 all use time travel as a device, by a different method each time.
This website lists almost every single Time Travel movie ever made, from 1921's A Connecticut Yankee in King Arthur's Court up to 2009's Star Trek.
In The Butterfly Effect, the protagonist has mysterious periods of blacking out. As a young kid, he starts writing a journal describing his feelings. Years later, he finds out when he reads the journal before a blackout, he goes back in time to the period of the blackout. He quickly finds his time travel has a type of Psychic Nosebleed limitation.
In Millennium (And the film of the same name) the world is badly contaminated, so the government sends people to go backward in time, capturing everyone who was on a transport (plane, train, or ship) where all of the people on the transport were killed, or an event (a war, an attack, an explosion) where everyone in the area dies, and replacing them with cloned dead bodies so as not to change history. The problem is that once anyone goes to a particular time, no one can ever go back to anywhere during that period, the time period - an hour, two hours, whatever - is blacked out and unreachable. Visit a plane flying over the Atlantic Ocean for an hour and you can't go to Paris, New York or Antarctica at the same time later on.
In Tempest: A Novel the main super power so far is the ability to time travel. In the beginning, the time travel works more like an inverse of Ripple-Effect-Proof Memory where the main character goes back in time to an alternate past, but cannot change anything in the present. Actually time travel comes up but it involves Alternate Timelines and other complicated rules which have yet to be fully explained.
In Seven Days, the hero was the only one who could work the device reliably, and he could only go back seven days at a time.
In Time Trax, the method varied, but the rules were that you could only travel between two set time periods (The Present and The Future), and more than two trips in a lifetime are lethal.
In the original Star Trek, time travel required either a dangerous and complicated slingshot maneuver or a precision jump into the Donut of Forever or Mr. Atoz's Atavachron, but these days Trek characters can travel through time by spilling coffee on their tricorder. (Which is probably why Star Fleet now has a department of time travel cops staffed entirely by grim-jawed Men in Black, as seen in DS9.)
Note that this isn't just a Plot Tumor (though it is one of those too)- time travel really is getting much easier in-universe as technology advances. By the end of the 24th century, it's shown, Starfleet's temporal function is beginning to overtake its spacial one. This is a large part of why they went to Prequels after Voyager. Of course, the Plot Tumor in question being TIME TRAVEL, this helped not at all.
Time Travel is such an amusingly big thing in Star Trek that, in Star Trek Online, Section 31 are revealed to have a star system set up specifically for pulling off the "slingshot around a star" stunt with precise calibration.
The series Voyagers centered around a time traveler and a young boy who travel through time trying to fix things that went wrong in history.
Blackadder Back and Forth featured Blackadder and Baldrick traveling through time when Baldrick accidentally made a working time machine. Then they go back to Set Right What Once Went Wrong when they change history by accident. Then they [[Set Wrong What Once Went Right Make Wrong What Once Went Right so Blackadder could become king. For once, it worked. ]]
In the Role-Playing GameFeng Shui, a region of cross-time 'space' called the Netherworld allows characters to move between four different points in history (69 AD, 1850 AD, 1996 AD and 2056 AD). These junctures are fixed with relation to each other, treating the start of the campaign as zero-hour for all of them. So, if you enter the Netherworld in 1996, travel back to 69 AD, stay for six months and then return to '96, it will be six months later there, as well. A second use of phlebotinum states that only people who control powerful feng shui sites can actually change the future by changing the past; everyone else just sees history work itself around the change.
In the card game Chrononauts, the players are time travelers from various alternate futures, and are trying to change the timeline to match their own timeline's version of the "past" so that they can finally go home. Since all the alternate futures have conflicting versions of "history," and many of those conflicting versions require a specific outcome to World War II (Hitler was assassinated early and WW 2 was Japan vs. America, Hitler lived and D-Day failed so that Germany won WW 2, and a couple other variants), Hitler's Time Travel Exemption Act gets a real workout. There's an alternate victory condition in which players have to collect certain combinations of Mac Guffins of questionable historical importance, but that's for material gain, not timeline shenanigans. A third victory condition is to get hired by the local Time Police after fixing enough of other people's paradoxes.
In the "Timeline" mod series for the original Half Life, rogue scientists from Black Mesa have figured out how to use the dimensional portals to travel through time. Gordon Freeman is elisted to...
Episode 1: Stop the scientists, who have given the time travel technology to Nazi scientists, to keep them from controlling key moments in history and changing the timeline... then when that doesn't work, going back in time again to make sure the nuking of New Yorknever happened...
In the Flux Family Secrets series the titular family is responsible for maintaining the integrity of the past.
The Clumsies and The Clumsies 2: Butterfly Effect involve going back and fixing the effects of accidental time travel.
The main character in Time Hollow had to do quite a bit of time-travelling (via a spiffy little pen that "drew" windows in the timestream) in order to fix the effects of temporal meddling and get his parents and old life back.
Narbonic: the 'Dave Davenport is Unstuck in Time' arc codified the rules, and future story arcs used the same rules. Fiddling around puts you in an Alternate Timeline. To move through time requires all the eneregy of an entire universe, thus utterly destroying an alternate universe/timeline in the process. Since you are in an Alternate Timeline, you can indeed change the future or the past. Dave is able to stop smoking by never having started. He is also able to give his past self information that saves the lives of Helen and Artie, and avoids a Bad Future.
In Time Squad the characters have to constantly go back in time in order to stop goofups in the timeline (because time is like a rope and as it grows it becomes frayed). Hilarity Ensues when they encounter historical figures doing crazy things, such as Eli Whitney creating flesh-eating robots instead of the cotton gin, Ludwig von Beethoven becoming a wrestler instead of a composer, or  thinking that the answer to all of the country's problems is a giant ball of twine.
Works (other than time-travel stories) that feature Time Travel in a major way:
Anime and Manga
In Mirai Nikki, its use is so incredibly spoileriffic details can't be given. Let's just say its important. Yuno Gasai abuses THIS.
Pokémon 4Ever features a Celebi that inadvertently brings the young Professor Oak with it to the present day when escaping from a hunter.
In Puella Magi Madoka Magica, this turns out to be the main power of Homura. The entire series is the nth iteration of a time loop that started when Kyubey granted Homura's wish for the chance to save an already-dead Madoka.
Soukou no Strain used the theory of general relativity to drive the plot. Sara's own motivations to become a Reasoner are to meet up with her brother, because if he returns from military service in space after a couple of years in his time, she'll be long dead in hers. Ralph's motivations are explained by his being able to go back hundreds of years using the same theory.
The Suzumiya Haruhi stories/anime feature time travelers, most notably Mikuru. It gets important in a major way in the novels, which also push Mikuru from being the Neutral Female somewhat. They travel to 3 years ago, and Kyon is the goddamn John Smith! The 7th novel also circles around it, this time with a Mikuru from a week in the future, setting off events to inspire the future inventor of time-travel and set off events necessary to bring about her organization. Like by nailing a can to the ground to send a man to hospital so that he can meet his future wife, or by dropping a turtle into freezing water to teach the inventor of time-travel something.
For everyone who has questions, I present you this◊. If it even helps. Careful of spoilers.
Tsubasa: RESERVoir CHRoNiCLE later turns out to have used this, having hidden it among a bushel of jaunts to alternate universes, or "countries". One "country" turned out to be the main characters' homeland in the past. And our world, or one much like it, in the future.
The Yu-Gi-Oh! Tenth Anniversary Movie features Paradox, a time traveling villain who wishes to change the past, and Yusei goes through a time slip. During the course of the story, both Judai and Yusei travel to Yugi's time, and at a certain point the Crimson Dragon takes Yugi 30 minutes back in time.
Zipang, where a JMSDF destroyer somehow ends up at the Battle of Midway. It's actually much more interesting that it sounds.
Lets Lagoon where the fog surrounding the deserted island can cause things to travel through time.
It's revealed in Eureka Seven Ao that Scub Coral uses time travel to transport parts of itself away to a different time and space to avoid the 1st series's calamity known as "Limit of Questions". It eventually lead to the existence of Secrets, the entire series major antagonists, and begins Ao and Generation Bleu's quest to stop Scub Bursts from happening. The 1st series hero and heroine, Renton and Eureka, are time travellers in Ao's world.
JSA has featured the modern Starman, a severe schizophrenic with powerful gravity controlling abilities. He claims, and it's probably true, that he is from a future Legion of Superheroes, future in terms of the Legion's comic too since he's an adult and the Legion in its comic is composed entirely of teenagers. Starman is also a dimensional traveler, who made his original appearance in Kingdom Come by helping Superman try and contain the villains and anti-heroes; apparently he can travel through time and the multiverse through a combination of his powers and a map that's written into his costume.
Prior to 1985Superman could time travel under his own power but would arrive in the past completely invisible and intangible, unable to interact with the past in any way, avoiding the problems with this trope. After 1985, he was no longer powerful enough to time travel at all.
Not quite. He would be invisible and intangible only if he traveled to a period where he already existed, since he couldn't be in two places at the same, er, time. If he traveled to a time prior to his own birth, he was solid. However he still couldn't change the past.
In the 8th-century Japanese tale of Urashima Taro. Urashima Taro is a young fisherman who visits an undersea palace and stays there for three days. After returning home to his village, he finds himself three hundred years in the future, where he is long forgotten, his house in ruins, and his family long dead.
The concept of travelling backward in time is relatively more recent. The idea was hinted at in Samuel Madden's Memoirs of the Twentieth Century (1733), and told more explicitly in Alexander Veltman's Predki Kalimerosa: Aleksandr Filippovich Makedonskii (1836).
The Time Machine inspired 99% of the modern uses of the concept. The book used it to provide a present day frame story for a tour of the future.
Zits in Flight time travels continuously by going into different bodies.
Dragonriders of Pern: The earlier books used the newly-(re)discovered time-traveling ability of the dragons for several plot points. After the Big One (Lessa bringing the lost Weyrs back thorugh time with her) time travel was relegated to a Save The Day plot device.
Which had more to do with the detrimental effects of dragon-based time-slipping: first, simply making the jump required traveling through the sensory-deprivation hell that is "between" for extended periods far beyond the quick three-breaths referenced in early stories, and second being in two places at once had ever-increasing mental effects on the travelers in question...effects that were decidedly unhinging to the travelers and intensified drastically the closer they were spatially to an earlier incarnation. Lessa's jump some four hundred years into the past very nearly killed her from apoxia, and the one recorded time that an earlier version actually caught sight of a later time-traveling one (for a split second, and even that only as a shadow moving in darkness) left the earlier incarnation almost completely physically and mentally incapacitated for a good fifteen minutes.
A Tale of Time City by Diana Wynne Jones. Time City is "outside" normal time, using recycled time (hence very important/emotional moments get burned in and are seen as time ghosts both before and after the event). Time is divided into unstable eras to be visited with great caution (ours obviously) and stable eras that they trade information with. However, they only sell information about the (relative) past, no stock market sneak previews.
In 1632, the 20th century town of Grantville, WV, is dumped into the middle of Europe, during the Thirty Years' War. Beyond that transportation back over three centuries, though, there is no more time travel.
The various protagonists of Michael Moorcock's Jerry Cornelius stories time-travel more or less constantly - in fact with Jerry it's damn near involuntary.
Oswald Bastable is also subject to this kind of involuntary shifting between alternate histories.
The Thursday Next series features multiple versions of history within a single book, but only the reader and the (off-screen) timetravelers are aware of this fact.
In the novel Rant, Rant uses a form of time travel to become his own stepfather.
The Time Scout series is built around an Accident that caused time portals to open up between random times and places. The stories cluster around people who happen to go places for various reasons.
Toward the end of The Secrets of the Immortal Nicholas Flamel, the conflict becomes less about stopping bad things from happening in the present and more about traveling to the past and ensuring that things happen the way they did.
In Gery Greer and Bob Ruddick's Max and Me and the Time Machine and Max and Me and the Wild West a time machine that one of the juvenile protagonists bought at a garage sale sent them into the bodies of people from the past for a limited time.
Live Action Television
Doctor Who is normally a variant of Adventure Towns, as the Doctor firmly believes that the timeline should not be altered, although some stories are concerned with the Doctor trying to prevent somebody else from changing the timeline.
Though this only seems to apply when the audience know what history "should" be. The Doctor won't save Pompeii from burning or steer the Titanic clear of the iceberg, but will happily stop a volcano erupting on the planet Tharg or a spaceship hitting an asteroid and exploding. The question "but what if the volcano on Tharg is *supposed* to erupt and kill everyone" is never asked. Bellisario's Maxim may apply here. The Doctor mentions in the new series that he can tell the difference between an event that can be changed and a fixed point in time which can't.
In fact, Doctor Who has generally been somewhat shy of actually using Time Travel as part of the plot, rather than merely a way of delivering the characters to the Adventure Town of the week.
Until the Steven Moffat era. Moffat's episodes are well-known for incorporating Time Travel or temporal paradoxes as an integral part of their plots, and the season arcs in his two years as executive producer have both focused on issues associated with the Timey Wimey Ball.
Star Trek: Enterprise: One of its central premises was a "temporal cold war", in which bandits are going back in time and messing with the timeline. The rules and limitations of time travel are never explained to anyone at any time, so the writers had a license to Ass Pull.
Voyagers - this was the entire premise. The 'Voyagers' were charged to Set Right What Once Went Wrong - they used one gadget, the Omni (which looked rather like a large gold pocketwatch), both to travel and to figure out what was wrong and how to set it right.
Kamen Rider Den-O features a superhero that travels back through time on a passenger train, DenLiner. Fairly early on, it is established that he is a "singularity point" a person who is completely immune to changes in the time stream and thus especially qualified to battle time-traveling Monsters of the Week. Why the OTHER singularity point handy, Hana, doesn't do the job remains unexplained.
Heroes has the character Hiro, his time travelling basically set off the whole first series in an attempt to change the future, it's a lot harder than you imagine, apparently. Also in the second series, he travels back in time and creates the character he heard in his bedtime stories. Peter also is prone to time travel but less often.
LOST from season 3 on, but especially in season 5. In Lost time travel nothing can be changed and everything is one huge Stable Time Loop. Note that the first person who claimed that time could be changed was fatally shot by his own mother before he was born once he actually tried to.
Prehistoric Park: A group of people (lead by Nigel Marven) set up a safari park filled with prehistoric creatures by traveling to the past and capture the creatures themselves. The time traveling device itself is never discussed in depth but it is what made the whole thing possible.
Achron takes the prime mention here - a Real Time Strategy game whose plot and gameplay are both mostly about time travel.
In The Elder Scrolls IV Oblivion a book tells the story of a famous battle in which magical time-altering storms were coming on an area and a local nation which knew their workings used them to deploy troops favorably. So they got hours of killing in where their soldiers outnumbered the enemy, had men in place to sack castles when hours turned to days, etc.
Final Fantasy XI uses this in Wings of The Goddess to travel 20 years ago to the Crystal War, one of the largest wars in Vana'deil's history.
Super Robot Wars Reversal has this as the main plot, the main characters got sent off to the past due to the encounter with the Big Bad and had to decide whether to let the future stay stable, or change it by modifying the past (they picked the second).
The Ecco The Dolphin series is all about time travel. The second game's plot even centres around the time travelling in the first game screwing up the time stream.
Legacy of Kain: big part of the plot. Especially in Defiance where point of view jumps between two protagonists in different eras, culminating in them both travelling to the same era to finally meet.
The driving force behind the plot of Pokemon Mystery Dungeon Explorers is that the god of time is slowly losing his marbles, and time is screwing up royally as a result.
Shadow Hearts: Covenant. Kato's entire plan hinges on going back in time 100 years to eliminate certain individuals. When the plan is ultimately foiled, everyone gets to pick a time to travel to to live happily ever after. Karin ends up going back in time, meeting Yuri's dad and becoming his mum. So...yeah.
Prince of Persia. The Sands of Time trilogy features 6-10 seconds of time travel as the primary gameplay gimmick. The entire point of Prince of Persia: Warrior Within is to Set Right What Once Went Wrong thus pushing your character's Reset Button. There's even a moment in Prince of Persia: The Two Thrones wherein the prince decides not to use the reset button again and man up to his mistakes.
Warriors Orochi 3. A monstrous eight-headed beast called the Hydra kills most of the heroes. The few remaining survivors are aided by Kaguya, the moon princess from Tale of the Bamboo Cutter, who uses her ability to travel through time to rescue the heroes who died in the battle with the Hydra.
Sonic the Hedgehog '06 employed this trope annoyingly, to the degree that after a series of confusing time-jumps (one of which is to undo sonic dying), a small fire is blown out, thus erasing the whole sequence of events, time jumps and all. This renders it effectively an "It was all a dream" scenario.
In Darkfall 2: Lights Out, the protagonist stumbles into one of several time portals, and must move repeatedly back and forth in time to figure out what's happening and return to his own era.
The changing-the-past equivalent was used thrice in Ratchet & Clank: A Crack in Time, the first time to save Orvus from Dr. Nefarious, the second to defend Zanifar from the Agorians, and the third to prevent Azimuth from killing Ratchet. The main plot also centers around using the Great Clock to travel back to prevent larger incidents. In Nefarious' case, he wants to wrong all the rights in the universe. For Ratchet and Azimuth, it's going back to prevent the Lombaxes' banishment. Either use would screw over the universe and all of reality, though.
Radiant Historia not only deals with time travel, but parallel universes caused by making different choices at certain points in time.
Time Travel is used several times in the Command & Conquer: Red Alert series by various factions, trying to improve their fortunes (generally by removing key enemy figures, such as Hitler or Einstein). It never goes well; the first game kicks off when Hitler gets cut from history, leading to a WWII between the Allies and Stalin, while in the third, the various time-travel shenanigans throughout the series have accidentally turned tiny backwater Japan into the Empire of the Rising Sun, a(nother) superpower bent on world domination. Hilariously, the Emperor believes in the "inevitability of destiny", and has a serious Villainous Breakdown when he discovers the truth behind the Empire's existence.
Magical Girl Lyrical Nanoha As Portable: The Gears of Destiny features a Time Machine Lost Logia discovered by a brilliant scientist who is trying to restore a dying world. The scientist, being the well-meaning and sane kind, decides not to use it since it for his purposes since that would cause too many complications to the timestream. Unfortunately, his daughter Kyrie, who doesn't want her aging father to die without succeeding in his life's project, decides to use it to retrieve an Applied Phlebotinum that only existed at one point of a specific timeline, kicking off the plot of the game.
The basis of BlazBlue involves a Stable Time Loop that plays over and over, prompting certain organisations to destroy a god in such a way that breaks said loop, but only so much that it keeps the world mostly intact. By the end of the second game, time has (supposedly) returned to progressing in a linear fashion. However those now in control seek to destroy the world entirely.
By making use of the time differentials between dimensions in Duel Savior Destiny it's possible to make large jumps forward in time and, it is implied, backwards as well. It's treated as something you have to work around rather than make use of, however. Near the very end it is confirmed that yes, you can travel backwards. This is how Taiga saved Crea from a monster that was going to kill her.
The sixth season of Sonic For Hire involves this. Sonic tries to go back to time with the Epoch to make sure he doesn't squander his life away. However, characters have been stealing his time machine and now many plotholes have occurred.
The Adventures of Dr. McNinja features time travel (as well as dimensional travel) in several arcs, including "Doc Gets Rad" (Big Bad Sparklelord gets defeated by being trapped in a Stable Time Loop), "Army of One" (In a flashback, a time-travelling Chuck Goodrich tries briefly to stop Doc from being cloned by Ben Franklin II), and "Space Savers" (Yet another Chuck Goodrich travels back in time [and into another universe] to stop a space dinosaur invasion). Since the comic works on Rule Of Cool, the precise rules for how all this fits together are never clearly established.
Bob and George. Oh, lord, Bob and George. One of the recurring catchphrases shared by many characters is "I hate time travel". George even suffers a nervous breakdown when faced with having to use it.
The Dreamer features an odd case of time travel. Whenever the heroine falls asleep, she is transported to 18th century America in the middle of the American Revolution.
Dresden Codak has a major plot arch which revolves around time travelers from the future entering and later invading the present.
Earthsong features a particularly head-spinning variant that doesn't actually CHANGE TIME AT ALL.
Homestuck incorporates a lot of Time Travel in its plot points, especially with the Midnight Crew intermission, where every single member of The Felt had a special ability related to manipulating time or alternate timelines. Within the main story, Dave (as the Knight of Time) has the ability to accelerate or reverse time around him. Alternate Future Dave becomes a minor character, but he is doomed to die since he's not part of the alpha timeline.
In Hanna-Barbera's video series The Greatest Adventure: Stories from the Bible, three young adult archaeologists find a door that takes them back to Biblical times. (Good thing the portal has random entrances and exits scattered through time, allowing one to cover thousands of years of Biblical history in a few weeks.)
Similarly the twin anime series Superbook and The Flying House are built around regular time travel into stories from the bible.
Time Squad involves time travel in almost every episode, as its name implies.
little known film Our Friend, Martin in which teens visit Martin Luther King at several points in his life and then bring him to their time, only to find doing so changes their timeline to one where his civil rights speeches and protests never happened (since he wasn't there to make them because he was in the future) so he must return home to restore the original timeline.
The WABAC Machine sends Mr. Peabody and Sherman back to assist historical figures in their quest for immortality.
The Powerpuff Girls episode "Speed Demon" has the girls racing for home so fast they go fifty years into the future and see that the world has been subjugated by their arch-foe Him because they weren't around to stop him as they went through time.
In the Al Brodax Popeye cartoons, Professor O.G. Wottashnozzle uses Popeye as a guinea pig for his time machine, which posits him and the others as historical figures.
Narrator: But where is he going, Professor?
Professor: I don't know. We take pot luck.
Works of fiction that occasionally call on this trope:
Anime and Manga
The Android Saga from Dragon Ball Z is kicked off by the arrival of Trunks, a Future Badass who owns both Frieza and King Cold when they come to Earth to seek revenge on Goku before revealing himself to be the son of Vegeta and Bulma. He's traveled back in time because the future he came from is a Bad Future where human civilization has been destroyed by the Androids and he wants to prevent that future from coming to pass by making sure that Goku doesn't die from the heart disease that he picked up on Planet Yadrat. He's only half successful because while Goku does survive the heart disease, he's out of action for the good part of the saga, leaving Trunks and the rest of the Z team to battle the Androids. Then an even more dangerous villain arrives from a third timeline...
Time travel is specifically taboo in the Sailor Moon universe, and it's the job of Sailor Pluto to guard the gate of time and make sure no one uses it. That said, The sailor soldiers (Chibi-usa especially) make occasional trips between the 20th and 30th centuries.
Yu-Gi-Oh 5Ds gives us the Infinity Device, which is capable of creating wormholes, useable for time travel. Illiaster intends to use the device to further their own schemes in guiding history on the correct path.
Mahou Sensei Negima, with the (multiple in the manga, one in the original series) version(s) of the Cassiopeia, which is (literally) a watch that allows you to time-travel.
On the other hand it also mentions that because of impatient building contractors with time machines, the great Cathedral of Chalesm was replaced by another building before it was ever built, thus making any pictures of it very, very valuable, blank, or both.
In The Restaurant at the End of the Universe, Marvin the Paranoid Android is abandoned for most of the lifespan of the universe due to time travel. It is later stated that, due to his various time travel incidents, Marvin is several times older than the Universe itself.
The novelization of the 1998 Merlin series implies that Lancelot came from a place in the future, or in a possible future, and was brought from it to the time of Arthur, by Merlin, to act as the king's champion. When Merlin first arrives there, they seem to have heard of him, though they never bring up what the history of their land says about it all.
Live Action Television
Babylon 5 called upon time travel in a few key episodes.
Lois and Clark had a few time travel episodes that included Time Machine author H. G. Wells.
LOST hinted mildly at time disparity in season 2, flirted with time travel in season 3, and took the full plunge by the end of season 4.
Nick Arcade had a Time Travel board where the player (Mikey) moves between the past and the future of his own neighborhood.
Power Rangers occasionally calls on this, even outside the Time Force season. Mighty Morphin' had a couple of trips back to the wild west era and the quest for the Zeo Crystals. SPD team had two separate time travel eps so they and the Dino Thunder Rangers could each visit the other team's home turf. Cam did the Kid from the Future thing on his quest to become Sixth Ranger, and Carter got the chance to repeat a day and save the lives of his teammates.
In season 2 of Roswell, Max travels back in time after everyone but he and Liz dies, in order to persuade past-Liz to break up with past-Max and make him get together with Tess. It's very silly and involves mariachis.
Stargate SG-1 had several episodes involving time travel—"1969" when they travel back to said year due to Stargate mishaps, Groundhog Day Loop episode "Window of Opportunity", "2010" showing a possible future where everyone is sterilized, "It's Good To Be King" with prophecies from the Ancient time-travelling puddle jumper, season-8 finale "Moebius" involving the same jumper and a twisted Time Loop (to be expected given the name), and season 10 Grand Finale "Unending".
1. Travelling through a wormhole that intersects with a solar flare causing the wormhole's course to alter sending the matter in transit back to the either the dialing Stargate, the destination Stargate or another Stargate altogether.
2. Using a time machine built by the Ancients to either get an area of a galaxy stuck in an ever repeating loop, or a Puddle Jumper with a time machine component that can only jump in jumps of 100+
3. Although not time travel persay, but, Asgard time dilation fields can be reversed to the time when the field was created.
Supernatural - Sufficiently powerful beings (e.g, angels) are capable of time travel, though it's not used often and changing the past was supposedly impossible until the Screw Destiny at the end of season five. In season six, Balthazar rewrites history by saving the Titanic; the incarnation of Fate, already pretty pissed at the main characters for putting her out of a job, draws the line at changing the past and coerces Castiel and Balthazar into restoring the timeline.
In Fallout 2 there is a random encounter, which sends you back to the prequels vault 13, where you break the water chip. Thus making you responsible for the events at the beginning of Fallout.
One of the missions in Osu! Tatakae! Ouendan involves being called by Cleopatra in Ancient Egypt to cheer on her helping her workers to build a Pyramid in 10 days so she can use its magic to get more beautiful and greet her lover Marc Antony properly.
Likewise, in Elite Beat Agents, one of the missions involve travelling back in time (by purpose) to Florence in the 15th Century, to help Leonardo Da Vinci win the heart of Mona Lisa and eventually create his masterpiece of painting.
The Legend of Zelda: Skyward Sword heavily features this mechanic in the Lanayru area: By hitting Timeshift Stones, Link can return an area in a certain radius from the stone to how it was in the past, also reviving any creatures whose remains lie in the area. So basically you can travel through time by walking into or out of the area of effect.
Time travel also plays a substantial role in the main story; among other things, the finale take place ages before most of the characters were even born, and Impa is escorting Zelda around the surface at the exact same time her older self is continuing to monitor the Imprisoned.
In Kingdom Hearts II, Pete's sheer nostalgia for the good old days when he was just a boat captain somehow opens a portal into Disney Castle's past. Unfortunately, his actions weaken the Castle's protection in the present, allowing Maleficent and The Heartless to invade. Merlin conjures the protagonists a magic door that lets them follow Pete and enter "Timeless River", a level-wide homage to early Disney. Everything is Deliberately Monochrome, and the present-day characters (except present Pete, who actually interacts with his past self) find themselves in their old outfits. Time travel isn't brought up again until 3D, where it's crucial to the plot of the entire game.
The current mega-arc of Irregular Webcomic! has massive time travellings done by many many characters in many many themes. This might be a Gambit Roulette on the part of the author to resurrect himself and Screw Destiny after he got killed by himself in the future and becomes Death of Going Back in Time And Killing Yourself and is suppose to go back and kill himself to continue the Stable Time Loop. Also, Leonardo da Vinci is a time traveller, is British, and made deals with Deaths. Did I mention that TARDIS also exist, and being used by the pirates and British navy crews (the latter owns it (?)), with the theme sets in 18th century? Yeah, it's that weird.
In Genius The Transgression, time travel is possible, but it's almost never a good idea. There's an entire section devoted to time travel and results thereof.
In Girls In Space, whenever the girls find the Earth, it is a different time period. They have no control over which time period has appeared.
The Life of Nob T. Mouse has Memory Lane, an area of space that allows people to see the past as if it is playing out before them. It's used on occasion to jog peoples' memories.
In Fafnir The Dragon, the use of this to prevent a post apocalyptic future is what drives the plot of the first chapter.
The Global Guardians PBEM Universe features the Warlord, a Powered Armor-wearing villain from the future. He didn't like the way things were going in his time, so he came back to change them. Every story featuring him involves him trying to change some historical event to fit his own whims.
Near the end of the Skyrim arc of WAOA, a few members of the group went back in time to learn Alduin's weakness. However, they bungled up Alduin's previous defeat at the hands of a group of heroes and turned Skyrim's present into a world ruled by Alduin. However, they got the weakness from a tablet left behind and used it to fix the error they made.
This was a primary plot-point in season 3 of Red vs Blue. During which Church is apparently blown into the past from the bomb that was placed in his gut, while the rest of the members of Red and Blue teams were blown into the future. Church escapes the past by having the computer Gary use his power to create a time machine so that he can go forward in time and stop any of this from ever happening. It turns out that time travel never actually played a role in this though, and that all of Church's experiences in the past were actually him being tortured by Gary (who is really Wyoming's AI Gamma)and the others just being blown away by the blast. Though all of this wasn't put in place until it was retconned by Burnie Burns so that it fit with the later story lines of the Recollection and Project Freelancer.
In "A Sitch in Time", a three part episode of Kim Possible, all three of the above plots are used. In the end, it turns out that time travel had been responsible for even the initial complication that got the plot rolling (Kim's sidekick moving to Norway) but all was undone by the end.
In Futurama, the crew of the Planet Express Ship gets sent back in time to 1947 Earth, and becomes the crashed alien spacecraft at Roswell, New Mexico. Fry does "the nasty in the pasty" and becomes his own grandfather, and Bender's head ends up buried in the desert for 1053 years, in a parody of the Star Trek: The Next Generation episode "Time's Arrow". ("What was it like being stuck in that hole for a thousand years?" "I was enjoying it - until you guys showed up!")
In a recent the episode "The Late Philip J. Fry.", this was taken to the extreme where Fry, Bender, and the Proffesor get into a time machine that only goes forward in time, causing them to keep going ahead in time looking for one that goes back, until eventually due to accidents and jerkassness, they went so far ahead in time they go through to the end of the universe, then another universe that's just the same is made in its place, then when they get to their time, an accident forced them to do the same a second time, where they came in about 10 feet over themselves before they went forward in time, they obviously dropped down and killed them, and took their place in that similar universe.
The Venture Brothers parodied this in Escape to the House of Mummies, Part 2 (there was no part 1), where the situation became increasingly ridiculous as they traveled around time, leading to Caligula, Sigmund Freud, Edgar Allan Poe, and two Brocks launching an assault.
Gargoyles had a magic item called The Phoenix Gate that could be used for time travel. Trouble was, it couldn't be used to change the past. Fate would simply conspire against anyone who tried to.
Of course our magnificent bastard villain, is still badass enough to still make his fortune using it.
Justice League had quite a few time travel stories, including one entire season that involved parallel universes and a stable but horrifying time loop that would result in a civil war between the world's governments and the world's superheroes. But it was all a Kansas City Shuffle by Brainiac-infected Lex Luthor; the time travel stuff wasn't real, just a red herring.
In "Melty", Lilo makes a fool of herself in front of her love interest, Keoni, and uses Jumba's time machine to go back to the past and change it. However, a side effect of the machine is that something (in a classic Ray Bradbury Butterfly effect) changes in each time line (which usually goes horribly bad). In the end, Lilo learnes a valuable Fantastic Aesop of literally not dwelling into the past.
In "Skip", Lilo and Stitch capture an experiment that is able to travel ten years into the future. In the first ten year travel, a seventeen (and shall I say HOT!) Lilo finds out that she has missed out on seven years of her life. When she goes another ten years in the future, everyting is hell. The villain Hamsterviel has taken over the island and the planet, captured all the experiments, and has become king of the galactic federation. Lilo decides that she can't force herself to grow up too early and conventiantly sets the reset button on the experiment to go back to the present time. My personal opinion to this episode is: Why didn't Lilo and Stitch starve to death when all that time went by?
Argai The Prophecy plays with this quite a bit, even with an original twist on it: When a character is killed in a time not its own, he or she doesn't die, he just returns to his original time. It's the reason the heroes must defeat Queen Dark in 2075, and for Queen Dark to kill Argai in 1250.
For a series that is so focused on the dangers of advanced technology, Jonny Quest The Real Adventures notably only had one time travel episode, "The Edge of Yesterday," near the end of its run.
In the world of Wakfu, Time Travel is the only time related power the Time Master race of Xelors doesn't possess. The Big Bad has to go on a genocidal campaign that has lasted centuries to gather an absolutely massive amount of Wakfu and pump it into a powerful Amplifier Artifact to make a trip through time possible. And he still only manages to go back twenty minutes.
The entire final season of The Smurfs was about time travel, coupled with Failure Is the Only Option as the Smurfs end up in one time period (and/or geographical location) after another.
The Young Justice episode Bloodlines is all about Bart Allen a.k.a. Impulse trying to prevent a Bad Future and the after effects are really confusing. In the future everything has become destroyed and covered in ash with only Impulse and the villian of the episode in sight. When Impulse changes the future the only thing that changes is that the villain was no longer a major threat in the past and doesn't have scars, but somehow despite changing that little the villian can still remember the old timeline.
Danger Mouse and Penfold find a grandfather clock that takes them through time in "The Hickory Dickory Dock Dilemma" (returning in "The Clock Strikes Back"). Has a nod to Doctor Who.
Special Mention Goes To:
Mentioned in the end, since this series uses (and spoofs) every single trope listed above:
Larry Niven's Hanville Svetz series of time travel short stories, collected in Flight of The Horse - where time travel is impossible in the real world, and every excursion that the protagonist makes is into a parallel, fantasy world that then directly affects his own. The reason for the jaunts? Well, the Secretary General of the UN in the series is a little mentally retarded, and the protagonist is sent back in time to recover animals that the SG has seen in recovered children's books. You see, they don't exist in the heavily polluted future... to the extent that, in one story where the proliferation of cars did not take place due to time meddling, one of the supporting characters has to breathe exhaust fumes from a internal-combustion car to stay alive. As is the case with most of Niven's work - it's all scientifically justifiable using the science known at the time of authorship.