Sane, 20, becomes the club's fifth signing of the summer and reinforces City in wide areas as he can play on either side of the attack.
The Germany international said working with new manager Pep Guardiola proved a big incentive for choosing a move to Etihad Stadium.
"I feel really good, I’m happy to be here and happy that it has all worked out. Now I can settle here in Manchester and play for City," he told the club's website.
One of the reasons I decided to join City was Pep Guardiola; he convinced me to come here and that I can continue to progress. I know I will learn a lot under him and now I can take the next step in my career.
“I followed Pep at Barcelona and at Bayern where he achieved a lot and worked really well with young players. I think he can make me a more complete player. I think I will need a bit of time to start with because it’s another league and there’s a different style of play but I think I can adapt very quickly.”
Guardiola: Sane will be a fantastic asset who excites fans
The forward scored nine goals in 42 appearances for Schalke last season and went to Euro 2016 with Germany, making an appearance in the semi-final defeat to France.
Guardiola believes City have a huge talent on their hands, having watched the player up close last season as Bayern Munich manager.
“He is a special talent, and an exciting player that I think our supporters will enjoy watching," he said.
He has great technical ability, is comfortable on the ball and there is much to admire about the way he plays football. He is fast, enjoys creating chances for others, works hard for the team and scores goals, too. He also has a very positive attitude and will fit in well with our squad.
“At 20, he is still developing as a player but he is already a member of the German national side and I think he will be a fantastic asset to our Club. We are very happy he decided to join us.”
Sane already settling in
Even before confirmation of his transfer broke on Tuesday morning, Sane had been getting to know his new team-mates, being pictured with striker Sergio Aguero.
How Sane will fit in at City
The capture of the youngster makes City's squad look even stronger after the signings of Spanish forward Nolito and German midfielder Ilkay Gundogan. The club have also signed promising Ukrainian youngster Alexander Zinchenko and Australian midfielder Aaron Mooy.
Guardiola certainly has plenty of options at his disposal - NB we have included John Stones as his £50m move from Everton appears close.