But the A-lister got into a bit of bother at one venue, The Tron pub near the Royal Mile, when members of his ten-strong party were refused entry due to their lack of ID.
Grant and his friends were hoping to see US comedian Eddie Pepitone, who, according to a source at the venue, "was a bit gutted that Hugh Grant had been turned away".
Speaking to The Herald, the unnamed source added: "There was no harm done, Hugh was an absolute gentlemen and his friends were very lovely and they just walked off. He was just wanting to see a comedy show and the bouncers did recognise him. But it was all because a couple of his friends didn't have ID."
For coverage of this year's Edinburgh Fringe and International Festival, visit Whatsonstage.com/Scotland