Icelanders use the traditional Nordic naming system, which includes the last name that is comprised of their father’s (or mother’s) first name with the addition of -dottir or -son.
Iceland has a few peculiarities when it comes to names. First of all, Icelanders don't have family names. Secondly, one cannot take up the spouse's last name upon marriage. Thirdly, when naming a child, one has to stick to a limited list of names.
Unlike most other Western countries Icelanders do not use family names but use a patronymic or matronymic reference. One's name reflects the immediate father or mother and does not refer to the person's historic family lineage. Although Iceland shares a common cultural heritage with the Scandinavian countries of Norway, Denmark and Sweden, it is only Iceland who continues to use their traditional naming system formerly common in all of Scandinavia.
The last name of a male Icelanders therefore usually ends in the suffix -son (son) and that of female Icelanders in -dottir (daughter). For example, Iceland's former president is Olafur Ragnar Grimsson, his first name is Olafur Ragnar and his father's first name was Grimur. Olafur's daughters are thus Guthrun Tinna Olafsdottir and Svanhildur Dalla Olafsdottir.
Another option Icelandic law provides is to use both parents' name as last name. For instance, musician Orvar Thoreyjarson Smarason (Örvar, son of Thorey, son of Smari) and former mayor of Reykjavik Dagur Bergthoruson Eggertsson (Dagur, son of Bergthora, son of Eggert) chose this combination of matronym and patronym.
Before 1925, it was legal to adopt new family names; one Icelander making use of that possibility was Nobel Prize-winning author Halldór Laxness, born as Halldór Gudjónsson. Since 1925, one can only do so if one explicitly has a legal right to do so through inheritance. Also when it comes to first names, Iceland has a few characteristics. It is common to not name a newborn right away but wait for a certain period of time to get to know the child. Usually this period lasts for about three months, in some cases even longer. Until the naming, the child simply goes by stulka (girl) or strakur (boy).