Nagelsmann has agreed a five-year contract with the reigning European champions, who have agreed a €25m compensation package with Leipzig for the 33-year-old’s release.
Bayern’s current manager Hansi Flick announced last week that he will like to step down from his position at the end of the season amid links with the Germany national team manager’s job and an uneasy relationship with the Bavarians sporting director Hasan Salihamidzic.
Nagelsmann is widely regarded as one of the brightest young coaches in European football.
After cutting his managerial teeth at Hoffenheim at the age of 28 in 2016, Nagelsmann took over RB Leipzig in the summer of 2019 and led them to the Champions League semi-finals last season.
Bayern president Herbert Hainer told the club's official website: "Julian Nagelsmann stands for a new generation of trainers.
"Despite his young age, he has had an impressive career. We are convinced that with Julian Nagelsmann we will build on the great successes of the past few years.
"I would like to expressly thank Hansi Flick on behalf of FC Bayern. He took over our team in a difficult phase in 2019 and then won six titles, the seventh will hopefully follow soon. He will always have a place in the history books of FC Bayern. We wish him all the best for his future."
Future CEO Oliver Kahn added: “Julian's five-year contract alone shows how much he identifies with FC Bayern.
"I am convinced that we will shape the sporting future of FC Bayern very successfully together with Julian Nagelsmann.
"My thanks go to Hansi Flick. With great football, he has led FC Bayern to great success. Now we have fulfilled his wish to leave FC Bayern before the end of his contract. All the best, Hansi!"
