The Red Devils have released a statement revealing that the 50-year-old Everton boss will take charge at Old Trafford from July 1, having agreed to leave the Toffees when his contract expires at the end of the current campaign.
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"I am delighted that Sir Alex saw fit to recommend me for the job. I have great respect for everything he has done and for the football club.
"I know how hard it will be to follow the best manager ever, but the opportunity to manage Manchester United isn't something that comes around very often and I'm really looking forward to taking up the post next season."
The former Preston manager also spoke highly of Everton and the club's board and players, insisting he will strive to achieve a strong finish to the season at Goodison Park, before departing for Old Trafford.
"I have had a terrific job at Everton, with a tremendous chairman and board of directors and a great set of players," he added. "Between now and the end of the season, I will do everything in my power to make sure we finish as high as possible in the table.
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The Glazer family have also expressed their delight at Moyes' decision to accept the post at Old Trafford, with Joel Glazer confident the Scot will be able to continue Sir Alex's legacy of producing "flair football played by exciting, world class players".
Avie Glazer continued: "The search for a new manager has been very short. Alex was very clear with his recommendation and we are delighted that David has agreed to accept the job. He comes to us with excellent credentials and a strong track record.
"I know he is keen to get started and Ed [Woodward] will be sitting down with him as soon as the season is over to discuss plans for the summer and beyond."