Paul Pogba said "the time is right to go back to Old Trafford" after completing his world-record £89m transfer to Manchester United.
The 23-year-old midfielder returns after four years away, having left United for Juventus for £1.5m in 2012.
Pogba, who has signed a five-year deal, added: "This is the right club for me to achieve everything I hope to."
Manager Jose Mourinho said the France international could be "the heart of the club" for the next decade.
United will pay the Italian champions 105m euros for Pogba, plus five million euros (£4.5m) in performance-related bonuses, and other costs, including five million euros if Pogba signs a new contract.
The fee surpasses the £85m Real Madrid paid Tottenham for Wales forward Gareth Bale in 2013.
The announcement was made by United just after 00:30 BST on Tuesday, with a glossy video showing the player - with a red devil marked into his hair - emerging from the shadows.
It was followed by a series of sponsor videos, including one with English rapper and grime artist Stormzy, whose most successful song to date is 'Shut Up'.
- Read: 'An absolute monster of a player' - Ferdinand on Pogba
- Read: 'Pogba gives Man Utd fear factor' - MOTD analysis
- BBC Three: Pogba follows in footsteps of football's great returners
- Listen: Pogba, the story so far
Pogba, who helped hosts France reach the Euro 2016 final, has won the Serie A title in each of his four seasons at Juventus.
He is Mourinho's fourth signing at United, after Ivory Coast defender Eric Bailly, Swedish striker Zlatan Ibrahimovic and Armenia forward Henrikh Mkhitaryan.
It is the first time in 20 years an English club has set a world-record transfer fee, after Newcastle United paid £15m to sign England striker Alan Shearer from Blackburn Rovers.
Pogba joined United from French club Le Havre in 2009 as a 16-year-old, but made just seven appearances before his contract expired in July 2012.
He has made 178 appearances for Juventus, scoring 34 goals, and helped the club reach the 2015 Champions League final.
The Serie A side had offered him a new contract, while Real Madrid were also interested in signing him.
Pogba said his mother told him he would return to Old Trafford and that it was his "destiny".
Speaking to MUTV, he added: "It was like I'd just come back home. I just went for a holiday."
Pogba said he had spoken to Mourinho "a few times", as well as talking to France team-mates who had worked under the former Chelsea boss.
"He gave me energy, he gave me positivity," said Pogba. "They all told me he's the coach for me - he will make me improve a lot and make me work harder."
In a series of messages to Juventus supporters on social media, Pogba said: "Goodbye means nothing, it is the time we spent together that matters. I will bring with me a piece of you and leave you a little piece of me."
The summer's most protracted transfer
The move has rumbled on all summer, with the back pages of the newspapers featuring reports of deals done, deals delayed, counter-bids from big-club rivals, denials and intrigue over agent's fees.
At Mourinho's first Old Trafford news conference on 5 July, he said he wanted to sign four players and had secured three. Many believed the fourth was Pogba.
It has taken more than a month since then for the Portuguese to get his man.
On 9 July, United declared their interest but there were rumours the player preferred a move to Real Madrid.
Following's France's Euro 2016 final defeat by Portugal, Pogba went on a long holiday to the USA.
His agent Mino Raiola said a number of clubs were interested in his client and continued to deny a deal with United had been agreed.
On Saturday, Juventus boss Massimiliano Allegri summoned Pogba to training, but just a day later United announced the player would have a medical - and on Monday he arrived in Manchester.
But it took until the early hours of Tuesday for the deal to finally be announced, a delay Pogba called "annoying".
Writing on Instagram he said: "This transfer took so long because it was a huge decision and involved lots of thinking and work."
Why did Pogba leave?
Pogba claimed he left the club "frustrated" and "disgusted" because former manager Sir Alex Ferguson did not play him. He gives his full reasons in the video above.
Ferguson said he "distrusted" Raiola, and alleged an agreement with Juventus had already been made.
How good is Pogba?
- Former France striker Thierry Henry believes he can go on to become "one of the best in history" while fellow World Cup winner Patrick Vieira described him as an "extraordinary player" and said it was a "mistake" for Manchester City not to sign him from United
- United legend Ryan Giggs said Pogba "has got everything"
- France legend Zinedine Zidane thinks Pogba is a "complete player"
- Juventus coach Massimiliano Allegri said in May 2015 that Pogba "needs to stop showboating in midfield" and he makes "avoidable" mistakes
- Arsenal manager Arsene Wenger claims Pogba is in a "maturing period" and will not find his best form until the age of 26
- Former England striker Gary Lineker asked during Euro 2016 if he was "the world's most overrated player"
- Ex-United midfielder Paul Scholes had questioned Pogba's £89m valuation, saying he was "nowhere worth that kind of money yet"
United took the unusual step of announcing the deal at 00:36 BST on Tuesday.
The club say that is when the deal was done and they did not want to hold the information in case the news was broken elsewhere.
It has been noted the time would coincide with morning commuters in the Far East, where United say they have 300 million supporters, and early evening in North America, the world's most lucrative commercial market.
"That is a little bit of a red herring," said Kevin Roberts, co-founder of SportBusiness Group.
"There doesn't seem to be any commercial advantage to timing an announcement in either of those markets ahead of the UK.
"In south-east Asia it would have generated a whole load of social media traffic. In North America there would be a massive amount of excitement amongst football fans generally. But while they have the information there is very little they can do to monetise it."
What did Juventus say?
The Italian giants thanked Pogba for "four years of triumphs lived together", which included nine trophies in total.
They released a statement 'Merci Paul', which said Pogba would remain "etched in the collective imagination".
It also said the club had made 72.6m euros (£61.9m) profit from the deal, after various deductions including wages and "related charges".
Match of the Day presenter and former England striker Gary Lineker tweeted: "The signing of Pogba by Man Utd is a watershed moment for the Premier League. First time a huge foreign star in his prime has come to England."
He added the wealth of the Premier League meant "more of the greats will come here and not just to Barca and Real Madrid".
When challenged about other big names such as Sergio Aguero, Alexis Sanchez, Yaya Toure and Angel di Maria, Lineker responded: "Either Barca cast-offs, not genuine superstars or built their reputation here."
Earlier, former Manchester United and England defender Gary Neville had said: "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."
BBC Sport's Simon Stone
"A lot is being made of the timing of Manchester United's announcement that they had officially signed Paul Pogba. The official release came at 00:35 BST.
"United are adamant the timing was solely to do with when the deal was completed.
"But whilst the timing was not great in the UK, it was perfect for the USA, where it was early Monday evening, and the Far East, where it was peak commuter time.
"Last month, United visited China, where they estimate they have more than 100 million fans. In the Asia-Pacific region it is more than 300 million.
"With the club heading for annual revenues over £500m, the days when club announcements were timed to assist local media are well and truly over."
- Under-20 World Cup winner and player of the tournament (2013)
- 4 Serie A titles (2013, 2014, 2015, 2016)
- 2 Italian Cups (2015, 2016)
- 2 Italian Super Cups (2014, 2015)
Most recent five times the world transfer record was broken
Hernan Crespo's £35.5m move to Lazio from Italian rivals Parma in 2000 was the last time a club other than Real Madrid broke the world record for a football transfer fee.
- 2013 - Gareth Bale £86m (Tottenham Hotspur to Real Madrid)
- 2009 - Cristiano Ronaldo £80m (Manchester United to Real Madrid)
- 2009 - Kaka £56m (AC Milan to Real Madrid)
- 2001 - Zinedine Zidane £46m (Juventus to Real Madrid)
- 2000 - Luis Figo £37m (Barcelona to Real Madrid)
My Manchester United XI
How do you think Man Utd should line-up for the 2016/17 season? Pick your XI - and then share it with your friends.
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