The German 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.’
It is reported the 29-year-old has signed a four-year deal at the Bernabeu until the summer of 2026.
His contract with the LaLiga side will be worth in the region of £315,000-per-week.
He will also net a signing on fee that will take his wage significantly north of that figure.
Sources close to the deal have also indicated that Rudiger’s release clause at Madrid will be €400m (£342m).
The eye-watering financial package is worth significantly more than the £230,000-per-week deal Chelsea were offering him to stay at Stamford Bridge.
However, Rudiger turned down bigger offers from elsewhere in Europe to sign for Real, who were the defender’s first-choice destination ever since talks opened in January.
Manchester United, Barcelona, Paris Saint-Germain and Juventus were all interested in Rudiger.
Rudiger won the Champions League last season having been brought back into the side by Thomas Tuchel following the sacking of Frank Lampard.
In five years at Stamford Bridge, the Germany international also won the FA Cup, Europa League, UEFA Super Cup and FIFA Club World Cup.
He joined the Blues from Roma for £29m in 2017 when the west London club were managed by Antonio Conte.