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Years ago there was rumours of a baby that glowed somewhere in China. This was the first known quirk.
All sorts of quirks have been discovered since then, ranging to all sorts of things. Some left people with physical mutations, some quirks are mind-related, some are power-ups. Really, there's all kinds out there.
Kids develop their quirk by four or five years old, getting either of their parent's quirks or some blend of both.
80% of the population develop quirks. Quirks have a very small chance of mutating beyond what their parent's quirks are if they have any.
But ever since quirks became more and more of a problem, society had struggles setting up rules for regulating them quickly. (quirk-ly? xD)
Hero was the title given to someone who stepped up and stopped villains who misused their quirks.
With every generation, more powerful quirks are born, and much the same with villains. Heroes courses are everywhere, but the biggest two schools are UA and Shinketsu.

Welcome to UA.