In most series set more than Twenty Minutes into the Future, the date will be an exact round number of years after the year the series was made. Something made in 1965 and set in the twenty-first century will be set in 2065. Something made in 1989 and set in the thirtieth century will be set in 2989.
This also applies to Time Travel, either forward or backward; from 1972, you can jump forward to 2072 or back to 1872.
There are two big exceptions to this. One is when they just use a nice round number by itself for the year; thus, all the TV shows and movies set in the year 2000 (or 1999, or 2001). The other is sequels to something that was set in Exty Years; for example, Star Trek: The Original Series came out from 1966 to 1969 and was mostly set from 2266 to 2269, but Star Trek: The Motion Picture came out in 1979 and was set in 2272, because it needed to be only a few years after the end of the series. Likewise for Star Trek: The Next Generation and its spin-offs, which took place almost a century after the beginning of the original series (2364 in TNG's Season 1) and continued in real-time thereafter, but originally premiered in 1987. Thus, when the titular ship of Star Trek: Voyager ended its, well, voyage in 2378, its last episode aired in 2001.
It should also be noted that characters, like real people, often round numbers off. Just because someone says something happened "1000 years ago," that doesn't mean it couldn't have happened 992 or 1038 years ago.
The trope's name comes from Homestar Runner's pronounciation of an X in "futuristic" dates, such as the setting of Mega Man in 200X and Metroid in 20X5. The series parodies this directly by setting their mock-anime mock-spinoff series in the year 20X6 (pronounced "Twenty Exty-Six"). That is, however, another trope.
The Human Popsicle, Sealed Evil in a Can, and Sealed Good in a Can have often been that way for Exty Years too. Similarly, it's the almost universal practice for the Class Reunion.
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Anime and Manga
The first Mobile Suit Gundam series is set in the year 0079 of the fictional Universal Century. The time it was released? 1979, of course. Several other Gundam series do this as well, for example, Gundam Wing (1995) is set in After Colony 195.
And don't forget Gundam00, set in 2307, made in 2007.
Sci-Fi anime Red Photon Zillion was aired in 1987. The story begins in -you guessed it- the year 2387.
Space Pirate Captain Harlock aired in 1977 and begins in the year 2977, yet somehow we'll still be using huge computers and landlines for the next millennium or so.
The plotline of Tenchi Muyo! has copious amounts of backstory going back for millenia; however, ALL the big important events seem to have happened a round number of years ago. To name just a few: Ryoko was imprisoned for 700 years, Ayeka and Sasami have been in cold sleep for just as long. Washu is 20,000 years old, and Kagato betrayed her 5,000 years ago.
The Crisis CrossoverDC One Million has an interesting variation on this; the future here is the 853rd century, exactly one million months after Action Comics #1, with each participating comic being written as Such-and-so #1,000,000.
Although not a "round" number, Judge Dredd is always set 122 years ahead of the present.
This is a round number from a different direction — the first issue came out in 1977, so 122 years after that was 2099, so as to set it just before the turn of the century.
The Sandman issue Men of Good Fortune used this across 6 centuries, beginning in 1389 and ending in 1989, the year it was written and published.
The Back to the Futuremovies start in 1985, go back thirty years to 1955, jump forward thirty years to 2015, and go all the way back a hundred years to 1885.
The original jump was at least justified in that Doc Brown selects 1955 as the date he invented time travel. Going forward to 2015, at the end of the movie, was given as a nice round number, after originally intending to go 25 years instead.
Though one could say that Doc Brown only selected these dates because the writers decided it would be Exty Years From Now.
It's implied that the Doc changes his mind to 30 years after the events of the movie, which presumably stick out in his mind since Marty made the 30-year jump. Those events had some pretty profound effects on the Doc's life.
And 1885 was a glitch in the already programmed 1985.
In Austin Powers: International Man of Mystery, Austin and Dr. Evil are frozen in 1967 and unfrozen in 1997, the year the movie was released. Similarly, Austin Powers: The Spy Who Shagged Me was released in 1999, but most of it takes place in 1969, after Austin and Dr. Evil travel back in time. Interestingly, the third movie, Austin Powers in Goldmember, does not use this "30 years" convention, although it does briefly involve time travel from 2002 to 1975 and back.
Star Trek IV: The Voyage Home is set in 2286 according to the official timeline. They travel back in time to 1986, the year the film was made, so they apparently went back exactly three hundred years. You'd think this would've at least been mentioned at some point in the actual film.
Star Trek, which was originally slated to come out in 2008, has most of its action taking place in 2258, a nicely round 250 years later
In Citizen Kane, the present year is 1941, young Kane was taken from his parents in 1871 (70 years ago) and he ran for governor in 1916 (25 years ago).
2057, a "future of science" film from the Discovery Channel, was released in 2007.
The film Metropolis, completed around 1926, is sometimes said to be set in 2026.
In Idiocracy, the main characters are frozen in 2005 and thawed in 2505.
Sleeper has Woody Allen getting thawed out of cryogenic sleep in 2173 - one of the doctors points out he'd been frozen for 200 years.
Used in The Matrix, though Morpheus acknowledges that it's only an estimate.
"You believe it to be the year 1999, when in reality it's more like 2199. I can't tell you exactly what year it is, because we honestly don't know."
For the interested, the actual time setting appears to be somewhere in the late 3rd millenium AD, based on the Architect's description of the history of the Matrix.
Zig-zagged in Looper: in dialogue, the time jump is described as "thirty years." DVD Bonus Content, on the other hand, says that the jump is thirty years, five months, and an exact number of days/hours/minutes/seconds.
1984 is a sort-of aversion; it's not a round number of years, but Orwell flipped the year he was writing in to get a "distant but chillingly near future" effect.
He wanted to title it Nineteen Forty Eight.
The movie version was not only shot in the real 1984, but every scene referencing a date was shot on the right date.
The Science Fiction book Frek and The Elixir takes place in 3003 (it was written in 2003).
Weirdly averted in the Lord Darcy mysteries: set in an Alternate History Europe, each story takes place in the same year when it was published in reality. This makes for some amusing dissonance, as Victorian-level technology and archaic royalist politics appear side by side with dates in the 1950s.
The futurist book 2081 was, of course, published in 1981.
The Crystal Maze Adventure Gamebook, published in 1990, was set in a futuristic version of the game show in 2090.
In Paths Not Taken, when Eamonn Mitchell hires Taylor to protect him from younger and older versions of himself, Taylor keeps straight which is which by estimating their ages, then thinking of them as "Eamonn 20", Eamonn 30", "Eamonn 50", and "Eamonn 60", not to mention "Eamonn 40" (his client).
Live Action TV
Time Trax featured a time machine which only could create a time jump or "arc" of 200 years, so they traveled from 2193 to (then present) 1993. As the time passed, so did the possible destination in the past, and we see the related actions of other future cops, so this is also an example of Meanwhile, in the Future.
Averted in the episode of The Twilight Zone titled "A Hundred Yards Over the Rim", in which a settler from 1847 is transported to the year 1961, 114 years in the future.
Lost in Space, particularly the original series, was set in 1997.
Amusingly, this didn't stop one or two set-on-Earth scenes from featuring horse-drawn carts and the like.
Doctor Who seems to exist for this trope. The number of times a story has been set exactly a whole number of decades (or centuries) in the past or future are too numerous to count. However, there are exceptions.
The vast majority of these exceptions are when the story in question was a historical piece; for instance, the Crusades, the French Revolution, the fall of Rome, the Battle of Hastings, etc;.
Some were just when the year was a round number in itself, though these are even fewer and further between. The Ark (original airdate: 1965) was set in the year 10 Million AD.
One definite subversion was in the serial "Trial of a Timelord" (original airdate: 1986) - in the first segment, no exact date was given; the second segment was explicitly stated as being set in 2379, however the third segment was set in 2986, which plays the trope straight.
"The End of the World" does label the date...as "5.5/Apple/26]]", and puts it 5 billion years into Earth's future. Where they've presumably put inanimate objects into the numeric system.
"The Waters of Mars" was meant to take place fifty years to the day after the airdate. It was out by six days.
Russ Davies seemed fond of round-numbered years, setting two of his stories in AD 200000 and 200100 respectively. He also "clarified" at some point that the year "5.5/Apple/26" was exactly AD 5000000000.
The various volumes of Lance Parkin's History of the Universe add more examples, by assuming that if the Doctor says "500 years in your future" or whatever, he means exactly that in the absence of any evidence to the contrary. Although if a date's in the millions, he'll go for the round number.
In Starstuff, Ingrid lives 30 years in Chris's future.
Averted in Lexx, where the prologue to the first movie takes place 2,008 years before the rest, and the gap between Seasons 2 and 3 is 4,332 years. The dialogue falls into this trope though, since both those numbers are given in their exact value once, then rounded to the nearest thousand each time it's mentioned later (and they're mentioned a lot).
The various collapses and catastrophes posited on Life After People seem to be colluding with the writers to comply with this trope, timing themselves for exactly 10, 20, 50, 100, 500 or 1000 years after humans vanish.
Sliders averted this consciously. According to the mythology, if our heroes missed the window to slide out of their current universe, they wouldn't score another opportunity for 29.7 years. Series creator Tracy Tormé was very steadfast on not rounding it off.
Although it's not quite the same thing, all the dates of the Entity's memories that can be traveled to in Chrono Trigger just so happen to be nice round numbers, at least for the centuries (600, 1000, and 2300 AD; 12,000 and — wait for it — 65,000,000 BC). The "Apocalypse", though, happens in the year 1999.
Can be attributed to Battlefield series with Battlefield 1942, later having a sequel set in the future entitled Battlefield 2142.
Some of the, well, battlefields on which players fight are also fairly obviously linked to real battles of Wolrd War II, and so is the backstory.
The reigning king of this trope is the Bridge at Remagen map, which is stated in its description to take place exactly two hundred years to the day after American forces crossed the same bridge as part of Operation Lumberjack in WWII.
Fable III takes place fifty years after Fable II, which in turn takes place 500 or 600 years after the firstFable game.
Robotron 2084 is a subversion; the titular date is 102 years after its 1982 release.
Homestar Runnerdid have a genuine example: Strong Bad read an email suggesting that he make a time capsule that would be opened "in at least X0 years". Naturally, this led to an Imagine Spot of Stinkoman finding the time capsule in the year 20X6, although that would have been exty-two years after the cartoon was made.
Sonic's plot takes place on January 9th, 4013 — exactly 2000 years after the film was released.
Every season of Futurama is set a thousand years after its original air-time. The exception is that apart from the very first episode "Space Pilot 3000", where Fry awoke on New Year's Eve 2999, most of the first season which was aired in 1999 was set in 3000, rather than 2999.