The Holland midfielder has signed a five-year contract and becomes the club's second summer signing after the arrival of Andre Schurrle.
He told the Chelsea website: "I'm very happy to be here at such a nice club with great players.
"I am really happy, the stadium is good and the training ground is fantastic.
"I know about all the players and will meet them on Monday. I expect a lot, I've watched matches and it's really nice to be here.
"I am a box-to-box player. I cover a lot of metres and can score a goal, but I also like to defend and think of myself as a real team player.
"I want to learn a lot, to develop myself. I want to play games and not only sit on the bench, that's what I hope for, to play in games at these great stadiums and hopefully be important for the team, whether it is from the bench or from the start."