The Icelandair pet policy is as follows:
July 2020: Due to the current situation, we cannot temporarily accept the requests to travel with animals in the aircraft cabin (for example, emotional support dogs). This is valid for flights until August 15, 2020, but subject to change depending on the situation.
Pets are not accepted in the cabin.
All pets transported on Icelandair travel in specially climate controlled transport areas in the cargo area of the aircraft.
Only dogs and cats are allowed in the hold area of the plane.
The following animal breeds are not accepted on Icelandair flights: Pit Bull Terrier, Staffordshire Bull Terrier, Tosa Inu and Dogo Argentino.
Taking a pet to a foreign country involves preparation and planning far in advance of your departure date. Requirements for pet travel to a foreign country are set by the destination country. If you are you traveling to another country and bringing along your pet you’ll need to meet the animal health requirements of the country you are visiting.
More information on the Icelandair pet policy