Gundogan currently injured but hopeful of being fit to play again in September.
Manchester City have signed German midfielder Ilkay Gundon has on a four-year deal.
Gundogan, 25, becomes the first City signing of new manager Pep Guardiola after completing a move for a reported £21 million.
The Germany international, who has been linked with a move to City for some time, is currently recovering from surgery after he dislocated his kneecap in training.
Gundogan, who in 2013 suffered a back injury which kept him out of action for 15 months, is hopeful of returning from his latest setback by September.
“I’m thrilled to have signed for Manchester City,” he said on City’s official website. “When I learned of City’s interest my heart was set on coming here and things have moved very quickly.
“I have loved my time with Borussia Dortmund and I would like to thank the club and the supporters for the five very happy years I spent there.
“They gave me a platform to play on and I became a better player because of the belief, patience and trust they showed in me.
“Now I have a new challenge and that is to achieve great things with City. The opportunity to work with a coach like Pep Guardiola is something I am really looking forward to and I am flattered to be the first signing the Club has made this summer.
“I can promise the City fans I will give everything to help us win titles both in England and the Champions League. These are exciting times.
“I have my best years ahead of me and I think we can achieve great things together. I can’t wait to begin this new chapter in my career.”
Gundogan is the second high-profile departure from Dortmund this summer following club captain Mats Hummels’ return to Bayern.
City’s director of football Txiki Begiristain added: “Ilkay is an excellent addition to our team and we are delighted to have him on board.
“He is an intelligent and versatile midfielder who can operate in a number of different roles. I think our supporters will enjoy his work ethic and technical ability and I believe he will be a very important player for us in the coming years.”
Dortmund sporting director Michael Zorc was gracious about the midfielder’s departure, thanking him for his service to the club.
“We would like to thank Ilkay Gundogan for five extraordinary years at Dortmund. BVB wishes him a speedy recovery and all the best for his future at Manchester City,” he said.
Described by his former coach at Dortmund, Jurgen Klopp, as an “intelligent and complete midfielder with numerous strengths”, Gundogan possesses a versatility and tireless work ethic that should help him thrive in the Premier League.