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Antonio Rudiger said it was a "proud" moment to be announced as Real Madrid's latest signing.
The defender has left Chelsea after five years, agreeing a four-year contract with the LaLiga and Champions League winners.
He moves on a free transfer as his Chelsea contract expires and is set to fortify Carlo Ancelotti's Madrid defence ahead of a fresh push for silverware next season.
Rudiger wrote on Instagram: "I'm proud to announce that I'll be joining @RealMadrid.
"I'm super excited for all the challenges ahead and can't wait to play my first games for this huge club."
His announcement came as Chelsea and Madrid both confirmed the deal was complete.
Rudiger will be officially presented by Madrid on June 20, the Spanish club said.
Rudiger enjoyed considerable success at Chelsea, winning the Champions League title with the Blues in the 2020-21 season when Thomas Tuchel's side denied Manchester City in the final.
He was an FA Cup winner in 2018 and helped the west London club land the Europa League title a year later.
Chelsea's Champions League defence in the 2021-22 campaign came to an end at the hands of Madrid in the quarter-finals, with Rudiger scoring as Thomas Tuchel's side won the second leg 3-2 in Spain but lost 5-4 on aggregate.