As reported by the BBC:
He will be replaced by Ted Verity, the current editor of the Mail on Sunday, who will have overall responsibility for both papers.
Announcing the move, Lord Rothermere, chairman of the Mail's owner DMGT, said Greig had been "outstanding".
Greig - who will now become a consultant editor - wished the newspaper "continued good fortune".
Eton-educated Greig joined the Daily Mail in 1983 as a junior reporter and has been its editor since 2018, when he replaced its long-running boss Paul Dacre.
He was previously editor of the Mail on Sunday and has also held editorships at society magazine Tatler and London's Evening Standard.
Announcing the changes on Wednesday, Lord Rothermere said Verity will become editor of Mail Newspapers, a role that includes overall responsibility for the Daily Mail and Mail on Sunday.
"Under Geordie's leadership, the Daily Mail has continued to inform and entertain millions of readers with the very best journalism, becoming the biggest-selling newspaper in the UK and winning multiple prizes for daily newspaper of the year," said Lord Rothermere.
"Geordie has been an outstanding editor of the Daily Mail and, before that the Mail on Sunday, and I thank him for his contribution over the last decade."
Greig said: "I am grateful to Lord Rothermere for 10 extraordinary years as editor of his newspapers. I thank everyone who has worked with me; my colleagues have been heroic and inspiring.
"I wish my successor Ted Verity good luck and also continued good fortune to the Mail.
"I look forward to new opportunities ahead and will bring the best of what I learnt from my years at the Mail on which I first joined in 1983 as its most junior reporter on the graveyard shift."
Earlier this week, Reuters reported other changes were being made to the newspaper group's leadership - with CEO of DMG Media Kevin Beatty retiring and being replaced by Richard Caccappolo, who is currently an executive at MailOnline.
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