Real Madrid president Florentino Perez told Eden Hazard the Santiago Bernabeu is his "home from today" as the Belgium international was presented in front of the club's supporters on Thursday evening.
The winger completed a €100 million (£88m/$112m) move from Chelsea last week after months of speculation linking him with a move to the Liga club.
Hazard is widely expected to fill the mantle left by Cristiano Ronaldo, who moved to Juventus last summer.
"We are Real Madrid, we want new triumphs," Perez said. "This club is hungry for victory and resistance to defeat. We're going to have a squad of great players and in front of it, Zinedine Zidane will be there.
"Today and now we welcome one of the best players in the world, Eden Hazard. Eden, you're where you wanted to be. Today the dream of your life is fulfilled.
"The Bernabeu is your home from today, here they will enjoy your quality, and soon you will notice that the best of this club are its partners and its fans.
"We know that Madrid was your big dream and for your quality and your work you are already here. We thank Chelsea that has collaborated to achieve your dream. These fans know that your football is special and different. Welcome to Real Madrid, welcome to your home."
Hazard's Chelsea contract was due to expire at the end of the 2019-20 campaign and he had repeatedly made clear his desire to join Real.
The 28-year-old addressed fans and media following Perez's introduction, speaking in French after explaining his Spanish is not yet up to scratch.
"This was my dream since I was a child," he said. "I wanted to play for Real Madrid. And now that I am here, I just want to enjoy it so thank you very much."
Hazard's former club, Chelsea, have been left in a tricky situation as they are currently unable to replace their star forward due to a FIFA-imposed transfer ban though they hope to overturn the ban on appeal.