Michael Jones was reported missing from the Leith area after having been last seen just before midnight on Tuesday.
He was last seen in Parliament Street wearing a Bob Marley t-shirt and a beanie hat. He was also carrying a rucksack.
He has also been described as 6ft tall and of medium build, with a red bushy beard and a Welsh accent.
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Officers said they are very concerned for Michael’s welfare and asked members of the public to help trace him.
Sergeant Richard Homewood said: “Michael has gone off for a few days on his own in the past but we are becoming increasingly concerned for his welfare.
“I would ask anyone who has seen him or knows where he is to get in touch.”
Anyone with information on Michael’s whereabouts is asked to call police on 101 quoting reference number 1277 of August 4.