The Future is typically 200-1000 years after the present time, but there are no real set limits, and Twenty Minutes into the Future has been popular at times. The Future differs from A Long Time Ago, in a Galaxy Far Far Away... by the presence of Earth—whether the show is set in San Francisco or whether Earth is a distant legend, there are always ties to Earth that make it significant in the show.
Most books of advice to aspiring authors insist that Space Opera should be set at least several thousand years in the future, based on just how much civilization would have to develop to make such things possible, but TV shows rarely go anything like that far ahead, partly to justify showing an Earth-based society that isn't so radically different that the viewers can't relate at all, but mostly because Sci-Fi Writers Have No Sense of Scale. (Although really, who can predict these things?) A relatively near-future Space Opera setting can be justified by having ancient civilizations already out in space and humanity a relative newcomer to the galactic stage, which has the bonus of being able to fit in expospeak as aliens explain what's going on to the ignorant human barbarians.
The Future is where much of "hard" science fiction takes place. The various Star Treks are set here, as are Babylon 5, Buck Rogers In The 25th Century, Firefly, you name it.
In The Future everything might get worse or better (The Future Will Be Better).
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Anime and Manga
ARIA takes place in the 24th Century on the planet Aqua, formerly known as Mars before terraforming transformed it into a water-covered paradise.
What many claim is the first example of TV anime ever, Astro Boy, has a VO at the start of most episodes declaring "...In the year 2000."
Guilty Crown is set in 2039, where a mysterious virus known as "Apocalypse Virus" is spreading. It does, however, blend over with traditional fantasy at times, as the protagonist is able to draw out people's personalities to use them as weapons. Some are more useful than others, though. This power also becomes the cause of some literal Fridge Brilliance, in episode 3.
All Gundam shows take place in the future, some further ahead than others.
One of the oneshots in the manga Robot Super Color Comic takes place in The Future. It's not known how far into the future however humans have long since began living in tall towers, one of the protagonists has apparently never seen wild birds, Sailor Fuku are no longer worn, and crepes have apparently changed in style (if they aren't nonexistent).
DC Comics's 1980's miniseries Conqueror Of The Barren Earth was a spinoff of The Warlord, set in the far future when the sun has begun to swell to its red giant stage.
In the DC Comics Crisis CrossoverDC One Million, half of the story takes place in the 853rd century (farther into the future than any other previous DC story, not counting the ones involving the End of Time). A highly detailed setting was created for this in which the entire solar system has gotten terraformed into Earthlike worlds, the whole human race has telepathic access to the Internet, and most amazing of all, descendants of DC's greatest heroes are STILL active!
ManTech is set in the future, though we don't know how far; the heroes' Sleeper Ship from Earth was discovered after having been adrift for an unknown length of time.
Marvel comics has Marvel2099 which is a Cyber Punk world inhabited by Legacy Character heroes. It's legal to sell dangerously addictive drugs and the Corrupt Corporate Executive types are even more ruthless. The future kind of sucks. But hey! There's also mass produced holograms and cool genetic engineering stuff!
The Warlord of Io graphic novel is set in the year 2853 A.D., shortly before the fabulous year 3000 of Rocket Robin Hood fame.
Any Marvel Comics story featuring time travelers like Kang or Cable is likely to give glimpses of their respective future worlds. Notably, since Marvel (Depending on the Writer, anyway) subscribes to a "branching timeline" theory of time travel, these futures need not agree with one another.
Idiocracy undermines the most popular ideas about the future. Life does not get any better, nor do we have any After the End scenario. Instead, selective breeding for stupidity has caused The Future to be actually much worse than the present, but not in a Mad Max way. The idea came from the '50's short story "The Marching Morons" by C.M. Kornbluth, which did not receive any credit.
Mr. Nobody takes place in 2092, which is fairly utopian in that senescence and death have been eliminated through medical technology. The titular protagonist is, at age 118, the last "mortal" human.
Isaac Asimov's Empire-Foundation novels take place in a galactic civilization so old that concepts like the laws of thermodynamics are considered to be "prehistoric" in origin. The origin of the human species is unknown; characters speculate on which part of the galaxy the oldest settlements are in, and some scientists propose that humans evolved independently on thousands of different worlds. Earth does eventually turn out to be the home of the human race, but this never becomes common knowledge. The first book takes place about 25,000 years from now
The Hyperion Cantos is set in the 2700s, after the farcaster system has allowed human habitation on many planets across the galaxy
David Brin's Uplift series features elements of the Stupid Newbie Humans version of 'near-future' (within this millennium) space opera.
The Osmerian Conflict is set about 400 years into the future. Although no explicit year is given when they switched from AD to SE, the new era (SE) has been in effect for about 310 years.
Doctor Who has frequently delved into Earth's future, all the way to a hundred trillion years on, during which time the universe faces imminent collapse. It tends to show it as fairly or sometimes decidedly Dystopian. (With Whoniversenot exactly set into stone, though, the series has shown some fairly contradictory visions of the future which include two contradictory accounts of Earth's ultimate destruction. For the record, it supposedly happens in seven million years time. Though other stories place it at five billion years from now.) Torchwood regular and Who expatriate Captain Jack Harkness comes from approximately the year 5000.
Retconned by the presence of the Time War. Apparently it caused a lot of really, really screwy stuff to happen to the timeline.
Firefly took place in the early 26th century (the pre-title montage indicates "2517 A.D.") in which Earth (known to everyone as Earth That Was) was stripped of all resources in a mass exodus to terraformed planets outside the solar system.
Pro Pinball: Timeshock! starts off in the "Distant Future" era, and requires traveling backwards in time as part of the game.
Xenon is set in some sort of unspecified future, with robot servants, flying people, and an all-controlling supercomputer.
BattleTech is set (for the most part) in the 30th-31st centuries AD. Not nearly as Grim Dark as 40k, but still a fairly unpleasant techno-feudal setting with a LOT of war. They do have peace at times, though.
But no range finders (only available to a select group of religious fundies), laser designators (only the Tau have them), or close-in air support, just endless ground-based war. If it ends on one planet, it begins on the other.
Paranoia is set in Year 214 of The Computernote though that's just because "Year 214" sounds nice, so who knows how long It's really been in power, with Troubleshooters packing laser pistols and rooting out Commie mutant traitorsnote due to a filing glitch, The Computer found some civil defense records from 1957 and mistook it for current intel.
Dead Space is set in a dystopian future near the beginning of the 26th century (the first game in the year 2508). Artificial gravity, huge space cities, telekinetic gadgets, moderate manipulation of time, and outer-galactic space travel have become commonplace; in fact, mankind now inhabits every solid mass in our own solar system (such as Saturn's moon, Titan) and several different parts of our own galaxy. However, there has already been at least one mass human extinction event due to exhaustion of natural resources on Earth, and so humanity had to resort to mining other planets. While mankind is actually on the path to recovery by the start of the game (planet mining has been going on for about 60 years by 2508), humanity is ruled over by the solar-system wide dictatorial EarthGov, and the only primary resistance is a religious cult. Oh, and of course, mankind has accidentally awakened horrifically violent and monstrously grotesque super zombies called Necromorphs that are systematically coming in to kill every human on every planet, and said cult is determined to help them succeed.
The Halo series takes place in mid-26th century, the first game occurring in 2552.
Hard Reset is set far enough into the future for robots to uprise in sufficiently vast numbers to force all humanity into a sole remaining city. This Bezoar City is a huge Layered Metropolis, where being far enough above the ground to hear wind whistle past you like in the mountains only means that you’re midway up the city.
Killzone is set in a dystopian future where the united human colonies (Earth itself is never involved in any of the games) are locked in battle against Helghast, the sole rebelling planet with great Nazi parallels to it.
Mass Effect takes place in A.D. 2183-2186. The relative closeness to the present day is a result of discovering Applied Phlebotinum on Mars, soon followed by the formation of the Systems Alliance and encountering the Citadel Council, both events resulting in humanity making technological leaps that otherwise wouldn't have come about for centuries, if at all.
It's still close enough to the present, though, that present-day pop culture and religion remain intact—mention is made, for instance, of "Old Yeller: The Centennial Remastered Edition".
Metroid uses an alternate calendar, but it is still pretty obvious future, with Powered Armor, space travel and more.
The backstory to Sid Meiers Alpha Centauri has Earth sending out the colony ship to Alpha Centauri in 2060, with a 40-year travel time that lands you on Planet in 2100. A normal game (assuming you don't go for the Zerg Rush) will typically last you into at least the 23rd century, with games lasting to the 24th and 25th hardly being pretty standard. During this time, you can research all kinds of cool technology, almost all of it being based on Hard Science well-documented to be possible as of the creation of the game (1997).
Space Empires is set sufficiently far into the future for there to be widespread space travel and various futuristic technology. In Space Empires IV, it is possible to construct whole Dyson Spheres!
Space Rangers begins in the year 3000, by which time the humans have spread out significantly into space and established friendly relations with most other races. Interestingly, The Federation is mostly averted: while each human planet has similar cultural norms and provides military ships for the Klissan War, the government is separate and there are plenty monarchies and dictatorships alongside planets wih democratic republican governments.
TimePilot: A video game where you started in a modern (1980s) fighter jet and started off against the first airplanes in the 30's. Different levels brought you to the present and then into the future, against space craft. If you survived past the last level, you reset back to the first, but with less ships and a faster enemy.
X Series is also set in the far future, though the level of advancement varied from game to game, especially if you take X Rebirth into account.
Xenosaga The human race switched to TC (Transend Christ) in 2510. The first game itself takes place in the year 4767, or A.D. 7277.
Xenogears takes place in A.D. 17276-17277, on a distant planet.
Ronin Galaxy: Although no specific date is given, the author refers to the setting as the "distant future." The reader is also clearly surrounded by Japanese culture, yet no actual nationalities or ethnic groups are named. Cecil wasn't born in Japan, he's from "The Moon" (AKA Japan in space.)
Nonpachyderm is set here, and although not explicitly stated, is around 200-250 years from the present day
The Orion's Arm universe is set over ten thousand years into the future. The intervening years are covered in the timeline in a fair amount of detail, and Earth is more or less a wildlife preserve/ historical landmark.
And since it's set ten thousand years into the future, everything is extremely odd.
In Aelan mythology from Ustal Naror islands, most events will happen in future: we life in the middle of the first of 58 chapters of era of eras of rafters (eras of eras are divided into chapters, and chapters are divided into ages lasting hundreds of years and eras lasting thousands of years)
Adventure Time supposedly takes place 1000+ years from our time, so it's somewhere in the 3000's.
Ćon Flux, possibly. No real time is indicated, and the setup itself seems to change episode to episode.
Futurama takes place in the year 3000. New York is in ruins (with a new city, called New New York, built over it), a destroyed pizzeria is part of a museum display, talking heads reside in jars, the Internet is an actual cyber world, robots run on alcohol, weird aliens are there- and killing oneself is as easy as going to a suicide machine, among others. Interestingly, average joe Fry seems to settle in the Future easily enough!
The Jetsons, with the cities floating over a generally unseen landscape. It's basically a transplanted sit-com which includes the Flying Car and Videophones, and where everything has a "space-", "moon-", "rocket-" or the like prefix, and people go to the moon and beyond as casually as we'd get on a plane.
Time Squad Nobody ever sets foot on Earth while the main cast is in their "present time", which is in the year 100,000,000. All of what we know about the grand future in this universe is that the Earth's nations finally obtained world peace and turned into one "super nation". There's also no war, no pollution, and bacon is good for your heart. Oh, and they have a military time travel force that keeps their utopian future from collapsing.