The 18-year-old left-back joins the club from Southampton in a £30 million deal and has signed a four-year contract - with the option for a fifth season - worth around £160,000 a week.
"I want to continue to progress my career and joining United is the ideal place for me to do that.
"I am looking forward to this new chapter in my life and also looking forward to learning from the world-class players and management at the club."
Shaw becomes United's second signing of the summer following the arrival of Ander Herrera from Athletic Bilbao on Thursday.
"Luke is a very talented young left-back with great potential," said United assistant manager Ryan Giggs.
"He has developed immensely during his time at Southampton and has all the attributes to become a top player.
"I am delighted he has joined Manchester United as I believe this is the right club for him to continue his development as a young player and eventually fulfil that promise."