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 am thrilled and excited to be joining Manchester United. I have been at Southampton since I was eight years old and I would like to thank them and the fans for everything they have done for me," Shaw told the Red Devils' official website.
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"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."