The France international has signed a five-year contract and the 105m Euros fee will potentially increase to £93m through several add-on clauses.
"I am delighted to rejoin United," said Pogba, who will wear the No 6 shirt. "It has always been a club with a special place in my heart and I am really looking forward to working with Jose Mourinho.
"I have thoroughly enjoyed my time at Juventus and have some fantastic memories of a great club with players that I count as friends.
"But I feel the time is right to go back to Old Trafford. I always enjoyed playing in front of the fans and can't wait to make my contribution to the team. This is the right club for me to achieve everything I hope to in the game."
The 23-year-old left United for Juventus in 2012 having made just seven senior appearances in all competitions for the club, all from the bench.
Sky sources understand Manchester United will not pay Juventus' agent's fee.
Mourinho said the signing of Pogba was key to his plans to revitalise the club and believes the player can join the list of Old Trafford greats.
"Paul is one of the best players in the world and will be a key part of the United team I want to build here for the future," he said.
"He is quick, strong, scores goals and reads the game better than many players much older than he is. At 23, he has the chance to make that position his own here over many years.
"He is young and will continue to improve; he has the chance to be at the heart of this club for the next decade and beyond."