The easiest way to travel to Tasiilaq is to fly from Reykjavik in Iceland. There are several airlines that fly to Kulusuk airport not far from Tasiilaq. From Kulusuk you can either take the local helicopter or try to find a local fisherman to take you by boat to Tasiilaq.
If you have booked your stay with a hotel or a campground, they will usually offer to pick you up at the Kulusuk airport with a boat. Otherwise, you can book your flight with Air Greenland.
There are international flights to Kulusuk, which is only a 10-minute helicopter ride from Tasiilaq. In the summer it is possible to get to the town by boat, in the winter it is more plausible to get there by snowmobile, dogsled or on skis.
