Chelsea Football Newsletter
The left-back is currently at the World Cup with Spain, and his transfer was made official on the morning of his country's first match of the tournament against Cape Verde.
Cucurella has signed a six-year contract at Real after Chelsea agreed a deal that could rise to £51.8m.
The 27-year-old is the first signing of Jose Mourinho's second spell in charge of Real.
Cucurella joined Chelsea for £63m from Brighton in 2022 and still had three years left on his deal at Stamford Bridge.
However, the defender has made no secret of his desire to return to Spain and has previously been linked with both Barcelona and Atletico Madrid.
In March, he said it would be "hard to turn down" an offer from Barcelona, who he played for earlier in his career.
Those comments came as Cucurella put his future at Chelsea in further doubt by criticising the club's decision to part ways with Enzo Maresca.
“We knew what Maresca wanted from us,” Cucurella said.
“Winning a title like the Club World Cup also helps, strengthens the bond, and you create great relationships during the celebrations. When a manager gives you that confidence and offers you a platform to fight for titles, you’d die for him.
“The moment Maresca left, it had a big impact on us. These are decisions taken by the club. If you asked me, I would not have made this decision.
"To make a change like that, the best thing is to wait until the end of the season. You would give everyone, the players and the new manager, time to get ready, have a full pre-season.
“The instability around the club comes from this, in a nutshell. We had a caretaker manager first, then a new manager, with new ideas and no time to work on them. It is what it is.”
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Real are also understood to be close to signing Bernardo Silva and Ibrahima Konate as free agents.
A deal for Inter Milan defender Denzel Dumfries is also said to be close as the squad is overhauled under Mourinho after two seasons without a major trophy for the club.