"Jose is quite simply the best manager in the game today," said United executive vice-chairman Ed Woodward.
Mourinho, 53, said: "To become Manchester United manager is a special honour in the game. It is a club known and admired throughout the world."
He said he had always "felt an affinity with Old Trafford" and claimed he has "always enjoyed a rapport with the United fans".
He added: "I'm looking forward to being their manager and enjoying their magnificent support in the coming years.
"There is a mystique and a romance about it which no other club can match."
Mourinho has been out of work since being sacked as boss of Premier League rivals Chelsea in December.
"His track record of success is ideal to take the club forward," said Woodward.
"He has won trophies and inspired players in countries across Europe and, of course, he knows the Premier League very well, having won three titles here."
In an interview with Mourinho said he is "in the right moment" to succeed at United, adding the "past three years" did not reflect the club's history.
"It is a giant club. It must be for the best managers," he said. "I prefer to forget the past three years and focus on the giant club I have in my hands.
"I want to win. What the players need to do is listen. I want to focus on the history of this giant club. I will give everything to go in the direction we all want."
"The most important thing is the players and my relationship with them."
Mourinho said United fans had shown him "empathy, no problems" on previous visits.
"I was pushed by that feeling to say things my clubs were not happy with, like when I won with Real Madrid," he added.
"Real Madrid were not happy with that."
The Portuguese's appointment followed three days of talks between his agent, Jorge Mendes, and senior United officials.
He has already been linked with a move for Sweden striker Zlatan Ibrahimovic, who played for him at Inter Milan.
Ibrahimovic, who has decided to leave Paris St-Germain, on Thursday that Mourinho "is the man" to bring United "back to the top".
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