He replaces Dutchman Louis van Van Gaal, who was dismissed on Monday, two days after the club's FA Cup win.
"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."
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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, told reporters on Thursday that Mourinho "is the man"

thank god

The doomed one.