After Nana Shimura's untimely demise at the hands of All For One, the superhero All Might (real name Toshinori Yagi) took her place as the wielder of the One For All Quirk. He became the most proficient user of the Quirk up to that point, with powers that included super strength, super speed, and invulnerability. Not only that, but All Might also took on the responsibility of being the #1 hero in the world and the so-called Symbol of Peace. He set out to be a hero everyone could count on by always winning against villains and performing his superhero work with a smile.
Before meeting Shimura, Yagi was Quirkless, but his desire to save people won Shimura over, as Izuku would with All Might years later. After gaining the One For All Quirk from Shimura, Yagi then attended U.A. High to train as a professional superhero (a big reason for its current status as the best school for training heroes). All Might also trained under the tough hero Gran Torino, a former friend of Shimura who possessed the super-speed Quirk known as Jet.
All Might even had his own sidekick during this time, the suit-wearing, straight-laced Mirai Sasaki, aka Sir Nighteye. Mirai had the Quirk Foresight, which (naturally) let him see into the future. Because of his Quirk, he was able to see that All Might would die in action if he didn't pass on his Quirk and call it a day.
