"The National Gallery of Iceland is the primary museum for the visual arts, and handles collecting, recording, preservation, documentation, presentation and research in the area of Icelandic visual art." From Article 5 of the Museum Act
The National Gallery of Iceland is the nation's art collection, and includes work by both Icelandic and foreign artists. The museum is open to anyone, and there is no admission fee for any of its exhibitions.
At other art museums, exhibition spaces and galleries members of the public may sometimes have to pay an admission fee. Admission is free at the Reykjavík Art Museum - including the Ásmundarsafn building, the Hafnarhús building and the Kjarvalsstaðir building.
There are generally admission-fee discounts for groups, children under a certain age, senior citizens and the disabled. There is also often one day of the week when admission is free, and some admission tickets are valid for more than one day. More information can be found on websites of the various museums and ticket agencies.