The 53-year-old Moyes signed a four-year contract with the Black Cats and was chairman Ellis Short's "first choice" to take over after Allardyce was named the new England manager this week.
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Moyes, who was sacked from Old Trafford in the immediate post-Alex Ferguson era and then had an unsuccessful stint in Spain with Real Sociedad, will be tasked with keeping perennial strugglers Sunderland in the Premier League for another season.
The Scot started his managerial career with Preston but really made a name for himself at Everton, before his ill-fated spell at Manchester United, where he was fired after less than a season in the 2013-2014 campaign.
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Moyes told the Sunderland website: "I am delighted to have joined Sunderland. I am relishing the challenge and excited by the opportunity.
"I have taken over a big British club, with a great support and I'm looking forward to working in the Premier League again.
"I look forward to continuing the good work done by Sam."
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