Midoriya Izuku was just a regular middle school student in a world where people with superpowers known as Quirks are the norm. However, he dreams of one day becoming a Hero, despite being bullied by his classmates for not having a Quirk. After being the only one to try and save his childhood friend Katsuki from a Villain, All Might, the world's greatest Hero transfers his own quirk, One For All, to Midoriya. The story follows Midoriya's entrance into Yuuei High School, a Hero Academy that aims to train and raise their students into great heroes.

Source: Wiki (edited)

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Sequel: Boku no Hero Academia (2017)
Other: Boku no Hero Academia (2016)
Side story: Vigilante: Boku no Hero Academia Illegals