Doncaster have made forward Dany N'Guessan their sixth summer signing.
Tottenham have confirmed that Romanian defender Vlad Chiriches has completed a move to Napoli.
The Croatia striker has signed a four-year contract with the Spanish champions and is a direct replacement for Chelsea-bound Diego Costa.
Mandzukic won the Champions League and back-to-back German titles in two years at the Allianz Arena, but his departure has been anticipated since Bayern agreed to sign Robert Lewandowski this summer.
The 28-year-old moves to Madrid on the back of a World Cup campaign which saw him score twice for Croatia before their elimination at the group stage.
Atletico's sporting director Jose Luis Caminero said: "The incorporation of Mario Mandzukic is great news because he has the features we were looking for to cover this position.
"He's a strong forward, with a great high ball control and a powerful shot. It allows us to start pressuring behind the rival from the beginning, because he works hard on playing the ball out of defence.
"He fits well with his attacking team-mates and is used to competing at the highest level. He's a winner, a player used to winning titles. He'll contribute very much."