The former Cameroon goalkeeper has joined from Inter Milan on a five-year contract worth an initial £43.8m.
Onana spent just one season at Inter Milan, having joined on a free transfer from Ajax Amsterdam last summer.
The 27-year-old made 41 appearances for Inter last season and kept eight clean sheets in 13 Champions League matches.
Onana is Manchester United's second summer signing after Mason Mount joined from Chelsea.
Onana comes in as a direct replacement for David De Gea, who recently left Old Trafford after 12 seasons.
Speaking to the club's official website after the move was confirmed, Onana said: “To join Manchester United is an incredible honour and I have worked hard all my life to get to this moment, overcoming many obstacles along the way.
“Walking out at Old Trafford to defend our goal and contribute to the team will be another amazing experience. This is the start of a new journey for me, with new teammates and new ambitions to fight for.
“Manchester United has a long history of incredible goalkeepers, and I will now give everything to create my own legacy in the coming years.
“I am excited by the opportunity to work again with Erik ten Hag, and I can’t wait to play my part in the success I know he is determined to deliver at this great football club.”
