The aircraft experienced steering difficulties and ended up on the grass at Aberdeen Airport while taxiing after landing yesterday.
Five people were in the plane, which was carrying out a routine patient transfer from Shetland to Aberdeen, but no-one was injured.
Flights were suspended for about two hours after the incident shortly before midday, with some services diverted to Edinburgh and Glasgow.
Passengers began boarding flights again and the runway re-opened, but delays were expected.
Passengers due to travel today were advised to check the airport website and to contact their airline.
The aircraft, a Beechcraft King Air 200, is operated by Farnborough-based Gama Aviation for the Scottish Ambulance Service.
The company has confirmed a full investigation will be carried out.
A spokeswoman said: "A Beechcraft King Air 200 aircraft operated on behalf of the Scottish Air Ambulance by Gama Aviation Ltd undertook a normal landing at Aberdeen Airport, following a routine patient transfer from Lerwick. Whilst entering the taxiway, the aircraft experienced steering difficulties and departed the runway on to the grass.
"The aircraft was carrying two flight crew members, one Scottish Air Ambulance paramedic, one patient and one patient escort, none of whom were injured.
"There was no damage to the aircraft. Gama Aviation is now undertaking a full investigation."
The Air Accidents Investigation Branch confirmed it would play a part in investigations into the incident.