After Nana Shimura’s untimely death at the hands of All For One, the superhero All Might (real name Toshinori Yagi) took her place as the wielder of One For All’s Quirk. He became the most proficient user of the Quirk to that point, with powers including super strength, super speed, and invulnerability. Not only that, 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 the hero everyone could count on, always defeating villains and doing his superhero job with a smile.
Before meeting Shimura, Yagi was quirkless, but his desire to save people captivated Shimura, as did Izuku and All Might many years later. After receiving the “One for All” quirk from Shimura, Yagi then enrolled at UA High to train as a professional superhero (the main reason for his current status as the best school for hero training). All Might also trained under the badass hero Gran. Torino, a former friend of Shimura who possessed a super-speed quirk known as Jet.
During this time, All Might even had his own suit-wearing buddy, straight-lace Mirai Sasaki, also known as Sir Nighteye. Mirai had a precognition quirk that (naturally) allowed him to see into the future. Because of his Quirk, he was able to see that All Might would die in battle unless he hung up the cloak and passed on his Quirk.