Pogba, 23, has signed a five-year contract with the option of a sixth season after United agreed to pay the Italian champions 105million euros (£89million) over the course of two years
That figure beats the estimated £85million that Real Madrid paid Tottenham for Gareth Bale in 2013 in the previous world-record deal
Juventus revealed United could owe them a further five million euros (£4.25million) if certain sporting targets are reached during the course of Pogba's contract.
Pogba left United four years ago for a nominal fee, and went on to win four Serie A titles and the Coppa Italia twice while his reputation grew - persuading Jose Mourinho to pay a colossal sum to bring him back to Old Trafford.
"I am delighted to rejoin United," Pogba said
"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."
Pogba's move has been one of the most protracted transfers of the summer, and there was a final tease before confirmation finally came in the early hours of Tuesday morning.
First, United posted a video on Twitter of a hooded figure in a club training top with the message "Are you ready? #POGBACK".
Moments later, Pogba's agent Mino Raiola tweeted "THE MAN IS UNITED".
And Pogba himself then followed up by bidding farewell to Juventus.
"Thanks @juventusfc and all Juventini
Goodbye means nothing, it is the time we spent together that matters ???" he wrote.
Only then did the official announcement come from the club
But United and their fans will hope Pogba proves to be worth the wait
"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," Mourinho said once the deal was official.
"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."
Pogba had arrived in Manchester on Monday to finalise the deal, travelling to the club's Aon Training Complex in Carrington in a red and black Chevrolet at lunchtime
Pogba is Mourinho's fourth signing of the summer, following Zlatan Ibrahimovic, Henrikh Mkhitarayan and Eric Bailly, but the size of the fee paid makes this the biggest statement of intent yet from the club's new manager
Speaking after Pogba jetted in, Red Devils defender Luke Shaw told MUTV: "After seeing that there was a good mood around the squad
"It is another player to fit in our strong squad, there'll be people fighting for places now and that's what you need at a top club like Man United to push us all on to win more trophies."
Former United defender Gary Neville, who was with the club when Pogba was first at United, told Sky Sports News on Monday: "I think it's important they sign players of stature and personality.
"And I think now with Zlatan and Pogba, there are players who will stick their shoulders out and have the strength of personality and character which I think United have lacked in the last two or three years.
"You think that (Ryan) Giggs has left, (Paul) Scholes has left, (Nemanja) Vidic, (Rio) Ferdinand, (Patrice) Evra, amongst others, who had huge personalties and were propping the club up in the changing room for many years.
"You need big shoulders and guts to play for Manchester United and I think the idea of Zlatan and of Pogba returning there is big on the personality front."
Ex-Valencia boss Neville added: "I think it is a big fee, but the Premier League was losing players left, right and centre to Spain, all the grade A players.
"We couldn't entice players to come here and we couldn't keep them when we had the best ones
At least now it looks like we are signing a player who would be regarded as a top signing of the summer.
"I think it's important United go back to signing players that actually fit the profile of the club, which is to play with dynamic strength, with power, with pace
I think some of the signings they have made this summer more fit that profile.
"And from my point of view, everything is too much money - not just Paul Pogba.
"United have spent quite a lot of money in the last two or three years without getting anywhere and I think it is important they made a statement in terms of this signing to show what they are.
"You say it's too much money, but for United, when they are pulling in the money they are, it is nothing for them."
Source : PA
Source: PA