France international Paul Pogba said the "time was right" to return to Old Trafford after completing his world-record £89 million transfer to United.
The 23-year-old midfielder returns after four years away, having left the club for Juventus for £1.5 million in 2012.
Pogba, who 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 Serie A champions 105m Euros for Pogba, plus 5m Euros (£4.5m) in performance-related bonuses and other costs.
It surpasses Gareth Bale's £85 million move to Real Madrid from Spurs in 2013 as the biggest transfer deal in the history of the sport.
The announcement was made shortly after midnight on Tuesday with a glossy video showing the player, with a red devil marked into his hair, emerging from the shadows.
Pogba, who helped France reach the Euro 2016 final, has won the Italian title in each of his four seasons at Juventus.
He is Mourinho's fourth summer signing at United, following in the footsteps of Eric Bailly, former Swedish striker Zlatan Ibrahimovic and Armenian Henrikh Mkhitaryan.
It will be the first time in 20 years that an English club has paid a world-record fee, Alan Shearer's £15 million move to Newcastle from Blackburn was the last.
Pogba joined United from Le Havre in 2009 as a 16-year-old but was left frustrated having made just a handful of first-team appearances before his contract expired in 2012.
Speaking to MUTV, he added:
"I just came back to Carrington, it was like I'd just come back home.
"I went on holiday after the European Championships and then the deal was worked on when I came back."
In a series of messages to Juventus supporters, 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 with a little piece of you." the Frenchman said in a leaving statement.
At Mourinho's first United press conference on 5 July, he said he wanted to sign four players and had secured three.
Many believed the fourth was Pogba.
But it has taken more than a month since then for the Portuguese to get his man.
Four days later, United officially declared their interest amid rumours the player preferred a move to Real Madrid.
On Saturday, Juve boss Massimilliano 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.
It took until the early hours of Tuesday BST for the deal to finally be completed in a transfer of a world star in his prime with unfinished business to return home.