Retro RPG: Gamma World
This is the 1992 Fourth Edition release of TSR's science fantasy role-playing game, originally designed by James M. Ward and Gary Jaquet, and first published by TSR in 1978.
Gamma World takes place in the mid-25th century, more than a century after a second nuclear war had destroyed human civilization.
Gamma World is a chaotic, dangerous environment that little resembles pre-apocalyptic Earth. The weapons unleashed during the final war were strong enough to alter coastlines, level cities, and leave large areas of land lethally radioactive. These future weapons bathed the surviving life of Earth in unspecified forms of radiation and biochemical agents, producing widespread, permanent mutations among humans, animals, and plants. As a result, fantastic mutations such as extra limbs, super strength, and psychic powers are relatively common. (Random tables of such improbable mutations are a hallmark of every edition of Gamma World.) Many animals and plants are sentient, semi-civilized species competing with surviving humans. Both humans and non-humans have lost most knowledge of the pre-war humans, whom Gamma World's inhabitants refer to as "the Ancients". The only group with significant knowledge of the Ancients are isolated robots and other artificial intelligences that survived the war—though these machines tend to be damaged, in ill-repair, or hostile to organic beings.
It's all a bit "Fallout", but long before Fallout was released.