Cristiano Ronaldo has joined Juventus on a four-year contract, bringing to an end his nine-year stay at Real Madrid.
The fee for the five-time Ballon d'Or winner is in the region of £105m, according to Sky sources.
A statement from Real Madrid on their official website said the Portuguese forward had asked to leave the club.
It read: "In response to the will and request expressed by the player Cristiano Ronaldo, he has agreed to transfer to Juventus.
"Real Madrid wants to express its gratitude to a player who has proved to be the best in the world and who has marked one of the brightest times in the history of our club and world football.
"Beyond the conquered titles, the trophies achieved and the triumphs achieved in the playing fields during these nine years, Cristiano Ronaldo has been an example of dedication, work, responsibility, talent and improvement.
"For Real Madrid Cristiano Ronaldo will always be one of his great symbols and a unique reference for the next generations. Real Madrid will always be your home."
Sky Sports reported earlier on Tuesday that Juventus president Andrea Agnelli was at Ronaldo's hotel in Greece to discuss the move.
The player's agent Jorge Mendes was understood to be meeting Real Madrid at the same time.
Ronaldo departs Real as the club's all-time top scorer, with an extraordinary 451 goals in 438 games.
He won four Champions Leagues, two La Liga titles and two Copa del Reys during his time at the Bernabeu.
In a letter posted on Real's website, Ronaldo confirmed he had asked to move on as "the time has come for a new cycle".
He also said the time he spent in the Spanish capital, following his then world-record £80m move from Manchester United, was "possibly the happiest of my life".
Cristiano Ronaldo's statement in full
These years in Real Madrid, and in this city of Madrid, have been possibly the happiest of my life.
I only have feelings of enormous gratitude for this club, for this hobby and for this city. I can only thank all of them for the love and affection I have received.
However, I believe that the time has come to open a new stage in my life and that is why I have asked the club to accept transferring me. I feel that way and I ask everyone, and especially our followers, to please understand me.
They have been absolutely wonderful for nine years. They have been nine unique years. It has been an exciting time for me, full of consideration but also hard because Real Madrid is of a very high demand, but I know very well that I will never forget that I have enjoyed football here in a unique way.
I have had fabulous teammates in the field and in the dressing room, I have felt the warmth of an incredible crowd and together we have won three Champions in a row and four Champions in five years. And with them also, on an individual level, I have the satisfaction of having won four Gold Balls and three Gold Boots. All during my time in this immense and extraordinary club.
Real Madrid has conquered my heart, and that of my family, and that is why more than ever I want to say thank you: thanks to the club, the President, the directors, my colleagues, all the technicians, doctors, physios and incredible workers that make everything work and that are pending every detail tirelessly.
Thank you infinitely once more to our fans and thanks also to Spanish Football. During these nine exciting years I have had great players in front of me. My respect and my recognition for all of them.
I have reflected a lot and I know that the time has come for a new cycle. I'm leaving but this shirt, this shield and the Santiago Bernabéu will continue to always feel as my own wherever I am.
Thanks to all and, of course, as I said that first time in our stadium nine years ago: Hala Madrid!