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.
Tsuyokute New Saga: The story begins within the climax of a very tragic take on the "heroes embark on a journey to defeat the Demon Lord" cliche; at this point most of humanity has been wiped off the face of the earth, and the heroes are on a suicide mission to kill the Demon Lord. They aren't expecting to survive or get back home. However, the last surviving hero, the leader Kail, manages to grab onto an artifact the Demon Lord seemed very concerned about before giving in to his wounds, and it sends him back in time by four years, before he lost everyone he loved. Knowing what's about to happen, he dedicates his life to preparing humanity for the coming disaster and making sure his family and friends survive this time.
In Future Diary, its use is so incredibly spoileriffic that details can't be given. Let's just say it's important. Yuno Gasai abuses it.
The Haruhi Suzumiya 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.
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.
A major theme and the focal point of Steins;Gate. There are multiple types of time machines and they all depend on the use of black holes. However, prototypes could only send back emails or some sort of electrical pulse because sending matter back in time causes it to turn into some jelly-like substance. Steins Gate uses the John Titor hoax from real life as if it were real, to create a plotline based on world lines and stable time loops.
Str.A.In.: Strategic Armored Infantry 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.
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.
Yu-Gi-Oh!: Bonds Beyond Time 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.
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.
The Android Saga 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...
Whis is revealed to have the ability to manipulate time, enough to go back by three minutes, anyway. It's debatable whether his ability is true time travel though, as it's implied that what he's doing is more along the lines of undoing the last 3 minutes across the entire universe, which is in some ways even more impressive.
Dragon Ball Super has this happen for its fourth arc, as Future Trunks once again travels to the past to escape the new Arc Villain, Black. This surprises Whis, who didn't think mortals were capable of developing the means to time travel... and makes Beerus angry, as apparently time travel is something even the gods are not supposed to do.
The Supreme Kais are able to time travel using "Time Rings", but they can only go to the future (and back again) to help themselves evaluate long-term consequences of mortal actions. Travelling to the past is absolutely forbidden: every time it happens, the timeline splits and an alternate reality is created (there are seemingly five in total, and the Supreme Kais have rings that allow them to jump from one to the other as well). Black has one of these Time Rings, which concerns Whis and Beerus, since only the Supreme Kais should ever be in possession of one of those.
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! 5D's 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.
This comes into play in the Wano arc of One Piece. Momonosuke's now-deceased mother, Kozuki Toki, had the powers of the Time-Time Fruit, which gave her the ability to send herself or other people forward in time. (Time traveling backwards is established to be impossible.) Toki was born in the far-distant past and repeatedly travelled forward in time until she met Momo's father, Oden, and settled down with him. When Kaido and Orochi took over the Wano country and killed Oden, Toki chose not to use her powers to save herself, but did use them to send Momonosuke and four of his retainers—Kin'emon, Kanjuro, Raizo, and Kiku—20 years into the future, to the current time of the story.