The France international was presented to the team's supporters ahead of Saturday afternoon's Premier League clash with Leeds United, where it was revealed that he will wear the number 19 shirt.
Man United had agreed a fee, thought to be worth £34m plus add-ons, with Real Madrid at the end of July, but the deal was held up due to various issues, meaning that he was not cleared to make his debut against Leeds.
Ole Gunnar Solskjaer's side have now officially signed the World Cup winner, who has signed a deal at Old Trafford until the summer of 2025.
"Manchester United is one of the most iconic clubs in world football and the chance to come here and play in the Premier League is something I couldn't turn down," Varane told the club's official website.
"There is a lot more that I want to achieve in my career and I know I'm joining a squad full of great players who will all have the same determination to win matches and trophies.
"Having spoken to the manager I can see how much progress has been made over the last few seasons and I now join a group ready to play at the highest level. I want to make an impact here and I will give everything to become part of this club's illustrious history."
Meanwhile, Solskjaer added: "I'm absolutely delighted that we have managed to secure one of the best defenders in the world over the last 10 years. Raphael is a proven winner who we have tracked over a long period of time and we know just how much of a dedicated professional he is.
"We have great depth of international defenders and he will add his immense skill set and leadership to that group. He is a unique defender with a rare combination of top-level attributes that I know will rub off on our younger players.
"He has won everything there is to win but I know that he is still determined to succeed, I can't wait to welcome him into the squad."
The 28-year-old, who could make his Man United debut against Southampton in the Premier League on August 22, made 360 appearances during his time with Real Madrid, winning three La Liga titles and four Champions League crowns. body check tags ::