Gamma World is set on an Earth which, centuries after some ill-defined global catastrophe, is populated with mutants of every mental and physical stripe, sentient animals and plants, insane malfunctioning robots and humans. The players are strongly encouraged to not take this very seriously. The RPG credits Lanier's Hiero books, Aldiss' Hothouse, and Andre Norton's Star Man's Son as influences (see Literature).
In the backstory of Warhammer 40,000, roughly around the 25th millennium, humanity's golden age was brought to a halt by "soulless" robots known as Men of Iron, and humanity descended into in-fighting until the God-Emperor managed to reunite much of humanity. However, the Imperium of man is now beset from the outside by aliens and demons and from the inside by mutants and heretics, and technological progress has effectively stagnated, with the "tech-priests" of the Adeptus Mechanicus content to seek the remnants of lost technology rather than invent new technology.
Humanity's golden age was brought to an end by the Horus Heresy, where fully half of the human race turned to worshiping demons and wiping out the other half. The God-Emperor was permanently injured in a lethal duel with their leader, Horus (in a way, the Emperor's son), to the point that he is only kept alive by an extremely complex life support device. Since then, it has stood on the brink of destruction for 10,000 years... and considering that it has been said there are flaws in the device beyond repair, it may be coming soon.
In 40K, where all emotions are shadowed in the Warp, faith is power - and the Emperor is the object of worship for an unbelievably huge and ridiculously fanatical state-enforced cult.
Known apocalypses in the Warhammer 40,000 'verse: The Age of Strife at the end of the Dark Age of Technology; the Horus Heresy at the end of the Great Crusade; the Age of Apostasy following the First Age of the Imperium; and, arguably, the Time of Ending, which is going on right now. After each one, humanity recovered; after each, the recovery was less complete, and society became worse. There are quite likely at least a few apocalypses that have simply not been named.
Now, now people, let's not be humano-centric here. There have been at least several other apocalypses involving other races going on: the most recent and significant must be the fall of the Eldar, although there is the apocalypse that wiped out the Old Ones as well. In fact look at any race (bar the Tau) and you'll find an apocalypse or two somewhere in their background (though with Chaos they were usually CAUSING them).
With the exception of the Eldar's apocalypse, which is widely believed to have caused Slaanesh, rather than the other way around.
Bar the Tau from having their own little apocalypse backstory? No, they were on the brink of wiping themselves out in a civil war before the mysterious Ethereal caste appeared in the eleventh hour to get them to work together. Not a mean feat considering their technological level at the time were at pikes and black powder cannons... The Orks have fallen from a previous state of civilisation, having inadvertently destroyed their Brain Boyz leaders and descended into happy anarchy.
BattleTech (Although the 'end' happens after humans have colonized space).
Battle Tech has had a few Apocalypses:
The fall of the Star League and the resulting 1st and 2nd Succession Wars which bombed the galaxy back to the Stone Age (barely 20th century tech, with how to make mecha and starships all but lost, though not how to repair them)... this was followed by 2 more wars lasting 300 years all together although these were low intensity conflicts because of the damage inflicted earlier.
The Word of Blake Jihad, a deliberate attempt to once again bomb the inner sphere into the dark ages, costing several TRILLION Lives over a 13 year-long war. Planetary saturation nuclear orbital strikes were common.
Exalted takes place after 3 different Ends, and is set at End of an Age. First was the Primordial War (named after the Primordials, the creators of the universe, who lose), involving the extinction of scores of civilizations and races; most of Creation was burned up by a sore loser's last act before surrendering. Then the First Age, was ended by the Usurpation. Lastly, a plague made by a ghost of a Solar killed in the Usurpation killed 90% of the population, and was followed by a invasion of The Fair Folk, who succeeded in unmaking half of Creation (by area).
The good news? You play EXALTS. They can do anything, including flattening the Fae and pushing them back to reclaim parts of Creation from the Wyld, and in the past they beat down the creators of the Gods themselves... Problem is the forces of the Underworld have Exalts too, and then there's that Great Curse. The potential to fix the world is there, it'll just take a lot of work, and the Exalts overcoming the effects of the Great Curse along with every possible threat to Creation... But hey, if anyone can pull it off, it's the Exalted
The Dark Sun campaign setting for Dungeons & Dragons was in its ancient past a typical Medieval European Fantasy world, but centuries of wizards abusing magic turned it into a blasted desert planet whose inhabitants have mostly turned to barbarism.
Game designers' early descriptions of what Dark Sun would be like actually referred to it as "the Forgotten Realms after they dropped the Bomb".
Palladium's flagship title, Rifts, takes place a couple centuries after the apocalypse. On midnight of the Winter Solstice in the year 2098, two nations in South America engaged in a brief exchange of nuclear weapons, killing several million people. The sudden death of so many people, combined with the mystic timing of the event, caused the Ley Lines crisscrossing the Earth to surge with a power unseen since the disappearance of Atlantis. This caused a number of weather anomalies across the planet which caused more deaths, fueling the ley lines even further, resulting in a chain reaction of death and increasing magical power. Eventually, the magical level rose to the point that the ley lines started becoming unstable, causing the eponymous Rifts, holes in time and space, to tear open and pour forth aliens and monsters, causing even more death. By the end, 60% - 80% of the Earth's population had been wiped out, leading to a Dark Age that lasted roughly a century, where humanity clawed its way out of the chaos and horrors caused by the Coming of the Rifts. In the main setting, Earth is now a dimensional hub where magic and technology exist side-by-side, sometimes peacefully, often violently. Humanity has regained a few footholds here and there, alongside aliens and other creatures who are as much victims of the Rifts as the natives, trying to eke out an existence on a world gone mad.
The supplement Chaos Earth takes place literally the day after it happens.
Paranoia is set after a nuclear war... or something... known as the Big Whoops, which ended with The Computer ruling over a huge population living underground in Alpha Complex which may be a dome city.
If the High Programmer book is to be believed it's San Francisco that's domed and underground, because a really big rock (or something) was going to hit the Earth, then commies took over the world
It's intentionally vague, but the main constant is not available at your security clearance, citizen.
According to the second-edition corebook, humanity retreated into the underground complexes to avoid a giant asteroid. Alpha Command, the computer that took care of day-to-day issues in human civilization, was damaged during the impact, and much of its data was lost. When it tried to boot up protocol files, it found an old '50s defense manual and assumed Communist sabotage, which it transmitted to the computers in charge of each complex - starting with Alpha Complex.
Earthdawn is set on Earth after a devastating invasion of other-dimensional Horrors wiped out most living things on the surface and mutated what was left. The Horrors (mostly) returned to their home plane after the level of magic dropped too low for them to stay, allowing the survivors to re-emerge from their retreats and begin to repopulate the planet.
Shadowrun is set in the future of Earthdawn, in 2070 our calendar, after a series of nasty magical cataclysms and wars (including a war between all the major continental European powers that ended with a massive nuclear airstrike on every nation's entire command structure by (presumably) England) and other disasters that created the politically-divided megacorporation-run Crapsack World it is.
The GURPS supplement Reign Of Steel depicted a post- Robot War setting where Earth is divided up and ruled by eighteen artificial intelligences and the human population is just 37 million, most of which are either in slave camps of one sort or another or are hunted like wild animals in the wilderness. The catchphrase: 'The war is over. The robots won.'
Also an invaluable resource for post-apocalyptic GURPS gaming is GURPS Y2K, which takes a long look at the fears of the turn of the millennium and the post-apocalyptic (or straight-out apocalyptic) scenarios that would develop.
The Day After Ragnarok, a setting book for both Savage Worlds, Hero System, and now FATE Core. The Nazis managed to pull off a summoning ritual that pulled the Midgard Serpent into our reality, but before it could fully manifest an American suicide team loaded the Trinity Device into a plane, rammed into the Serpent's eye, and detonated the bomb. The flailing around of a gargantuan serpent (whose head alone is 350 miles across) in its death throes while dripping magical radioactive snake venom from its fangs crushed most of western Europe and northern Africa.
Mutant Future is a close-as-you-can-get-it retroclone of post-apoc RPGs such as Gamma World using the Labyrinth Lord rules.
Aftermath! is an old Scavenger World game, and it's slim pickings since it's a few generations after the end. There's lots of scenarios for the setting, lots of little rules systems for simulating special cases, lots of genetically engineered life forms and Schizo Tech and you are as likely to die of starvation and exposure as violence.
Legend System's Hallow setting is built from the remains of a solar system, with floating islands powered by living engines (called angels) keeping everything running.
Apocalypse World takes place roughly fifty years after some sort of apocalypse. A suggested overarching goal for the characters is finding out what caused it and seeing if they can fix it, or at least move it from a Crapsack World to a Cosy Catastrophe.