7 Seeds is set who-knows-exactly-how-many-years after meteorites have crashed into and doomed the earth. The earth went into a mini ice-age and is only now starting to be inhabitable again, with evolution repeating itself. The only remaining humans on Earth are selected teenagers (and adult guides) who were frozen shortly before the apocalypse.
Attack on Titan is set in an unspecified future where humanity has been reduced to a handful of walled villages that are constantly under threat from roving giants that seek to eat them alive. The story follows the main defense force of the last city on earth as they seek to take the fight back to the monstrous Titans. This is later revealed to be a lie; while the island the main cast lives on is overrun with Titans, the rest of the world is made up of thriving human civilizations.
The Big O is set in a city which was built after an unknown event caused mass total amnesia. Everything else is believed to be barely-inhabited wasteland.
Blue Gender is set after the titular Blue has wreaked such havoc and destruction on modern society that a majority of humanity was forced to retreat into space and live on a station dubbed "Second Earth". Those who weren't that lucky live day after day in perpetual fear that the Blue will swoop down and kill them at any moment. Buildings lie abandoned and the Blue infest nearly every area of the Earth.
Blue Submarine No. 6 is set in after Dr. Zorndyke causes mass flooding on Earth and the remaining cities are getting ravaged by fish-human hybrids
Desert Punk. The apparent undertone of the series is that Humans Are Bastards, leading not just to the backstory disasters but also to the continued strife in the present.
In Dragon Ball Z, Future Trunks' timeline is this thanks to a pair of rampaging androids and why he comes to the main timeline to prevent it from happening there.
It also happened three centuries prior to the main storyline when King Piccolo nearly destroyed the world leaving only Master Mutaito and his pupils to fend themselves. Piccolo was eventually sealed and Earth began to rebuild.
In EL, the Megaro Earth Project is implied to be the last remaining city populated by humans after nuclear war devastated the Earth. At the end of the second OVA, it is revealed that the situation is far worse as only El and a single human male have survived, kept in a coma-like state by sentient AI hoping to find a way to breed them.
Ergo Proxy takes place after a "detonation of the methyl hydrate layer", and as a consequence, the remains of humanity live in giant domes protecting them from the outside world. Said calamity may or may not have been the direct result of nuclear war. Rapture looks suspiciously like those pillars of flame showing during end of the world flashbacks.
Luckily the original human inhabitants saw it coming, or at least responded in time and fled into space, leaving the Proxies behind to rebuild civilization. It didn't turn out very well.
Fist of the North Star is set in 199X, where the earth has been devastated by nuclear war and nearly all living things have gone extinct. Evil warlords and vicious gangs roam the land, killing and subjugating the weak. Thankfully, there are heroes ready to defend the people and make these villains pay, several of whom are badasses with a mastery of a mysterious but horrifying Pressure Point based martial art.
From the New World takes place in a distant future after something caused the decline of mankind.
Genma Wars took place in a distant future where mankind had long being conquered by a demonic tribe known as the Genma, who reduced it to hunter-gatherer level. Some ruins containing technology from modern times still exist, but nobody is able to operate it. Its revealed through time travel that this was possible because the Genma had infiltrated positions of power in the government and used the media to demoralize the populace, before developing genetically-engineered mutants and nuclear weapons to destroy the world, so they could rebuild it in their image.
Genesis Climber MOSPEADA (another show that became part of Robotech) has humanity on Earth reduced to a few small scattered pockets by the Inbit. However, the Inbit are seemingly uninterested in humanity's space colonies, which the latter take advantage of by turning Mars into a major military base for their efforts to retake their homeworld.
Getter Robo Armageddon as well as the Hien and Äḥ installments of the manga.
GUN×SWORD takes place on the prison planet of Endless Illusion after Earth has been destroyed.
In Gundam, several series have used this as a premise:
Gundam X, set 15 years after a whole lot of Colony Drops very nearly killed all life on Earth. The survivors have just come out of a 15 year long winter caused by dust blocking out the sun.
∀ Gundam, it's been a couple thousand years since the most destructive war in history was fought, and resulted in the complete annihilation of all civilization on Earth. Humanity had to re-develop from the Stone Age, and only now has the Earth's environment recovered from the damage.
Heat Guy J. Apparently nuclear war trashed civilization to the point that only seven widely-separated city-states remain in the world. To prevent a repeat, a watcher race called the Celestials maintain and judge each city. At least one city was deemed unfit and shut down. Beyond the cities are seemingly just a scatter of primitive tribes (the remnants of said city comprise one).
Hyper Police — the apocalypse brought most fantasy creatures to Earth (gods, oni, beastpeople including werewolves), and left humanity as a protected, endangered species.
ICE takes place in a world where all the men have died and Homosexual Reproduction not yet invented, so the series focuses on the remaining women trying to survive the end of the world.
Kemono Friends takes place on the ruins of the now seemingly abandoned Japari Park, which is full of Friends and Ceruleans but devoid of any human civilization except for Kaban. Just uncovering what's going on and what happened to everyone is one of the main focal points of the show.
Kiba no Tabishounin The Arms Peddler takes place on Earth after a cataclysmic war wiped out modern civilization where humans have to not only survive the wasteland but also defend themselves against raiders, bandits, slavers, mutants and supernatural creatures.
Part of Kurozuka takes place after the world has been devastated by unintentional nuclear holocaust (the nuclear powers were trying to destroy an asteroid, but accidentally targeted each other instead) and the resulting nuclear winter.
La Maison En Petits Cubes doesn't specify but appears to be set in a time after some sort of catastrophic sea level rise. An old man lives in a rooftop apartment in what was once a large city that has become inundated by flooding. The flooding has been gradual, as the man has been able to add room after room to the top of his house over the years, creating a tall tower. Only a few building peaks show on what was once a city.
Magic Knight Rayearth in the second half of the story, due to the death of the Pillar. Everyone in the country has been moved into a tiny castle surrounded by crumbling wasteland being eyed by foreign invaders.
Megazone 23: Long ago, the Earth's climate collapsed under the weight of humanity's abuse. Humanity left in generational ships, of which only one survived to return.
The world of the Monster Rancher anime is explicitly stated to be this. The human civilization which created the monsters was crippled by natural disasters, then the survivors fought an increasingly destructive war until Moo was created to be the ultimate weapon, except he decided to just destroy everything instead. The last few survivors barely managed to create the Phoenix and defeat Moo. By the time the series takes place, civilization has returned to a Medieval level with occasional Schizo Tech.
In the backstory of The Mysterious Cities of Gold, Mu and Atlantis fought a nuclear war that wiped out both empires (as well as most of the rest of humanity) and almost all of their uber technology.
Possibly Naruto. Between the severe levels of schizo tech, the bizarre structures in Rain country, and the wreckage that the Uchiha weapons stash appears to be located below, it's never been explicitly stated but it would explain a lot. The probable source of said apocalyptic event is most likely the Juubi. The wreckage in particular has an after the end feel to it.
Neo Human Casshern takes place in a time when Androids have taken over and a man who placed his mind in a superior android fight back against the androids with the help of his dog, Friender.
The series Casshern Sins that aired well over thirty years after the original is after the end — of the end, as now the androids society that conquered humanity is crumbling because of a disaster caused by this version of Casshern, who lost his memories and wanders the ruined Earth searching for answers to questions he doesn't know.
And the 2004 live action movie adaption: Casshern.
To a lesser degree (in that only about half of humanity is gone), Neon Genesis Evangelion. The goal of the protagonists is ostensibly to stop various people and things from finishing the job. Ostensibly.
This happens for real in Rebuild of Evangelion 3.0., which takes place 14 years after 2.0., where the Third Impact that occurred there wiped out most of humanity and apparently ravaged the planet even worse.
In Neppu Kairiku Bushiroad, a poison later known as "Shinobi" came to our world from space, after which despair took over, with the people being driven to a place known as "Kairiku". The only things left that can save humanity are a mysterious human weapon by the name of "Yagyu", and the holy weapon "Giga Road".
Now and Then, Here and There is a case where this is after and during, as Shu gets transported to a future where the Earth has dried up and it's orbiting a sun that's about to go nova.
Overman King Gainer has the cast believing that the Earth has healed from mankind's presence, and trying to reach Yapan, the home of their ancestors, from Siberia where a good portion of humanity went to live in Domes.
Panzer World Galient: Several millennia before the beginning of the show, the world of Arst was devastated by a terrible war. Most of the survivors left the planet as the rest started to rebuild it. Tens of centuries later Arst is inhabitable again, but their inhabitants technology has regressed to medieval levels.
Planzet is set in 2053 after a war with aliens has destroyed most major cities. Yet the army and Komoyo's school (in Tokyo anyway) are still running (though Taishi mentions that walking outside is risky because of bandits.)
Psyren is about a group of people that are being sent back and forth to what remained of Earth after the cataclysm that wiped out most of the existing lifeforms and transformed the surviving ones into dangerous mutants.
Rave Master. Technically, the world the story is set in is a parallel world created by the last surviving human after famine, drought, and plague destroyed the original world. The main villain, the only descendant of that sole survivor and, therefore, the only 'real' person, wants to go back to that world.
Sky Girls Although everything LOOKS nice and neet and clean it takes place after an alien invasion that has killed off nearly every man on Earth leaving girls to defend Earth should there be a second invasion. Possibly only 1 out of every 100 people left is male.
The story of Snow White and Seven Dwarfs takes place in a post-apocalyptic and isolated Tokyo (though it's implied that the rest of the world is fine), after meteors hit the city two years ago and destroying a fair amount of it.
Sonic the Hedgehog: The Movie: the strange fantasy world in this movie appears to be this, but it's never really explained. The creatures of this world appear to have rebuilt society and moved on... or they just don't mind abandoning tall infrastructures and New York City sized settlements.
Super Dimension Fortress Macross (one of the main shows to be incorporated into Robotech) has its penultimate arc end with most of the Earth devastated by a Zentradi fleet, with the surviving humans and their "cultured" Zentradi allies forced to rebuild civilization afterwards; Macross City is the new capital, built around the SDF-1, with homes, shops, schools, and offices situated in a barren Alaska, with other cities miles away (possibly around Canada and some of the USA, although it's not specified where each city is located). The creators of the franchise eventually wanted to show a much cheerier image of the post-apocalypse after the big victory, however, by having the capture of a Zentradi satellite factory helping humanity to not only rebuild, but to colonize the stars; as shown in Macross's many, many, manysequels, humanity and its allies has been mostly thriving in the many decades since the end of the original series, with multiple colonies and fleets spread across a good chunk of the Milky Way.
Sukasuka is set in a world where all the humans barring the main character are now extinct following the war between them and the god-like Visitors and the mysterious rise of the 17 Beasts who now plague the land and turned the surface world into a desert wasteland. The surviving races in the present day have now fled to and dwell on the floating islands where they're remotely safe from the Beasts, though there would be the occasional invasions certain Beast species wouldn't mind doing to some of the islands that are inhabited...
The world of Toriko was ravaged by war in the past. Then the legendary chef Acacia invited the world leaders to a full-course meal consisting of ingredients from the Gourmet World. After eating the main course called "GOD", the world leaders experienced enlightenment and worked together to bring about peace.
Trigun: An interesting variation. Turns out the current state of affairs came to be after humans ruined earth and then escaped into the universe. Then some drama happens among the crew, resulting in millions of people dying and the remainder being haphazardly dumped on the planet where the story is set.
Trinity Blood opens to a future where it's 900 years after an unnamed disaster nearly kills off almost all of mankind.
Tsubasa -RESERVoir CHRoNiCLE-, in Acid Tokyo, a Tokyo destroyed by acid rain where the survivors protect the eponymous reservoir, and, consequentially, Clow.
Played with when it's revealed that Acid Tokyo is the ancient past of Clow Country. Those ruins Syaoran was investigating at the beginning are the remains of the city's buildings.
Vampire Hunter D. After two ends, no less. First a nuclear war late in the 20th century or early in the 21st that reduced humanity back to the Medieval era, and the slow hubris of the vampire super civilization that followed five thousand years later, and lasted another five thousand years before the humans were truly back in charge, yet still underdogs in the distant frontiers.
It's easy to miss the fact that Vampire Knight is set in the far future, after the apocalypse, long enough after for things to have been rebuilt to about the same level. In a recent flashback, Juri tells a story of her adolescence to the young Yuuki; we've heard before that she was about three thousand years old, and the architecture and setting appears to be roughly modern day Japan, certainly within the last 30 years or so.
It's more likely that the setting of time everything takes place in is fictional. During another flashback where Juri reveals her first pregnancy, the setting here appears to be set around more ancient times again, with a horse carriage and old fashioned clothes. We don't know how long after the "Umbrella" bonus Juri and Haruka had their first child, and as most of the story centers around the Cross Academy school, Kuran's mansion and the Hunter Association's headquarters, we know very little about how the rest of the world looks like in terms of what era it is.
Violence Jack by Go Nagai is perhaps the first post-apocalyptic anime/manga. The series is set in the chaos after a massive earthquake and volcanic eruption leveled Japan and focuses on the survivors of the Kanto region, who are divided between the strong and the weak. It's noted for its over-the-top violence and brutality, and would serve as the inspiration for many violent anime and manga to come.
Wolf's Rain, though it is not initially clear, it is shown that the world is in the middle of a nuclear winter, and the few cities and towns huddled in domes that we see are one of the only bastions of humanity left.