From Vault 4 to Vault 76 and beyond
Every vault in the Fallout series is explained | Fallout Lore
[Ed. note: This post contains spoilers for Fallout Season 1, as well as information from the Fallout games.]
The Vault systems in the Fallout series are meant to be a utopia, a refuge away from the harsh apocalyptic wasteland and nuclear fire.
Of course, nothing is ever as it seems, and a little digging (Lucy's journey in Fallout Season 1 or playing to the end of Fallout 2) reveals that the Vault systems are actually a way to experiment on survivors. Some of the premises are so wild or impractical that it doesn't seem like an experiment at all, and many failed spectacularly. (According to series creator Tim Cain, the purpose was to test humanity's ability to travel through space, but this does not exist in any of the games so far.)