The 29-year-old had already announced he will be leaving Chelsea this summer after failing to agree an extension to his expiring contract.
Rudiger joined Chelsea from AS Roma in the summer of 2017 and won one FA Cup, the UEFA Champions League, UEFA Super Cup, FIFA Club World Cup and Europa League in his time at the club, while making 203 appearances and scoring 12 goals.
The Germany international has signed a four-year contract with the newly crowned European champions.
Writing on Twitter, Rudiger said: "I'm proud to announce that I'll be joining Real Madrid. I'm super excited for all the challenges ahead and can't wait to play my first games for this huge club."
"We bid farewell to Antonio Rudiger as he departs Stamford Bridge with his Chelsea contract expiring, bringing to a close his five trophy-filled years with the club," Chelsea's statement announcing Rudiger's exit read.
"Rudiger made 54 appearances in all competitions in his final campaign, all from the start, more than any other Chelsea player. He departs with 203 appearances and 12 goals to his name, in that time establishing himself as one of the most passionate characters to represent the club in recent memory.
"Chelsea Football Club thanks Toni for his contribution to our success over the last five years and wishes him well in his future career."
