Giggs in interim charge at United
The 40-year-old, who was player-coach under Moyes, will be in charge until a permanent replacement is appointed - with Holland boss Louis van Gaal the front-runner.
The club said in a statement: "Following the departure of David Moyes as manager, Manchester United has announced that Ryan Giggs, the club's most decorated player, will assume responsibility for the first team until a permanent appointment can be made.
"The club will make no further comment on this process until it is concluded."
United will try to move quickly to install a permanent replacement for Moyes. Van Gaal is the favourite to take over, with another potential candidate, Borussia Dortmund manager Jurgen Klopp, out of the picture.
United are determined not to make the same mistake as last summer when Moyes did not take over until the start of July, meaning a big chunk of the transfer window had already passed.
Van Gaal's credentials make it clear why he is a strong contender - he won the 1995 Champions League as coach of Ajax, has won Spanish league titles with Barcelona and reached the 2010 Champions League final as well as winning the Bundesliga with Bayern Munich.
He is also used to working with big-name players and attracting big signings - neither of which were part of Moyes' track record.
Van Gaal has also made it clear that he will be looking to move on from being Holland head coach after the World Cup.
Klopp has been touted as a candidate for the top jobs in European football, most recently as the next Barcelona head coach, but he wasted little time in ruling himself out of the Manchester United job.
The 46-year-old told The Guardian: "Man United is a great club and I feel very familiar with their wonderful fans. But my commitment to Borussia Dortmund and the people is not breakable."
Klopp, who agreed a new contract at the Bundesliga club until 2018 back in October, said last week that he was "still in love" with Dortmund.
Bookmakers have slashed the odds on Van Gaal from 8-1 to clear 11-4 favourite.
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