The incident happened at about 16:10 on Monday on Abercromby Place at its junction with Dundas Street.
A 73-year-old man was attempting to cross the road when he was struck by a silver-coloured saloon car, which then failed to stop.
It was last seen heading east along Abercromby Place. Police are appealing for witnesses.
The man sustained serious injuries and he was transported by ambulance to the Royal Infirmary of Edinburgh where he remains in a stable condition.
Sgt Iain Blain, of Police Scotland, said: "We are pursuing every line of inquiry in relation to this incident and we're eager to trace the driver of this vehicle.
"I urge anyone who can help identify this person to contact police immediately.
"Similarly, if you were in the area of Abercromby Place on Monday afternoon and saw the silver coloured saloon vehicle then please get in touch."
A pensioner is in hospital with serious injuries following a hit-and-run as he crossed an Edinburgh road.