Icelandic names differ from most Western family name systems by being patronymic (and sometimes matronymic) in that they reflect the immediate father (or mother) of the child and not the historic family lineage.
Iceland shares a common cultural heritage with the Scandinavian countries of Norway, Sweden, and Denmark and its crown dependency the Faroe Islands. Icelanders, however, unlike other Scandinavians, have continued to use their traditional name system, which was formerly used in all of Scandinavia although it has also been retained by a minority of the Faroese population. The Icelandic system does not use family names. A person's surname indicates the given name of the subject's father (patronymic) or in some cases mother (metronymic). (The words patronymic and metronymic derive from Greek pater, father, and meter, mother, + onuma, name)
Some family names exist in Iceland. Most of them are inherited from parents of foreign origin, but some are adopted. One notable Icelander who has an inherited family name is football star Eiður Smári Guðjohnsen. Before 1925, it was legal to adopt new family names; a notable Icelander who did so was the Nobel Prize-winning author Halldór Laxness. Since then, one cannot adopt a family name unless one has the legal right to do so due to inheritance.
Given names that have not been used in Iceland before have to be approved by the Icelandic Naming Committee before being used. They are accepted or rejected based on whether or not they can be easily incorporated into the Icelandic language. First, they must contain only letters found in the Icelandic alphabet. Second, names must be able to be declined (that is, modified according to their grammatical case).
The same practice is used for daughters. Jón Stefánsson's daughter Katrín would not have the last name Stefánsson; she would have the name Jónsdóttir. Again, the last name literally means "Jón's daughter" (Jóns + dóttir).
In some cases, an individual's surname is derived from his/her parent's middle name instead of the given name. For example, if Jón is the son of Hjálmar Arnar Vilhjálmsson he may either be named Jón Hjálmarsson (Jón, son of Hjálmar) or Jón Arnarsson (Jón, son of Arnar). The reason for this may be that the parent prefers to be called by his/her middle name instead of first, which is fairly common, or that the parent's middle name seems to fit the child's given name better.
In some cases where two people in the same social circle bear the same given name and the same father's name they are socially distinguished by their paternal grandfather's name. E.g. Jón Þórsson Bjarnarsonar (Jón, son of Þór, son of Bjarni) and Jón Þórsson Hallssonar (Jón, son of Þór, son of Hallur). This method is not common (as middle names are more commonly used), but such tracing of lineages can easily be seen in the Sagas.
In all of these cases, the convention is entirely the same: Fjalar, the son of Bryndís, will have the full name of Fjalar Bryndísarson (literally meaning "the son of Bryndís"). One notable Icelander with a metronymic name is football player Heiðar Helguson, Heiðar son of Helga. Another is Guðrún Eva Mínervudóttir, Guðrún Eva daughter of Mínerva. One medieval example is the poet Eilífr Goðrúnarson.
Some people have both a metronymic and a patronymic; one example is Dagur Bergþóruson Eggertsson, former mayor of Reykjavík.
Icelanders formally address others by their given names. For example, former prime minister Halldór Ásgrímsson would not be addressed as Ásgrímsson or Mr. Ásgrímsson by another Icelander; he would either be addressed only by his given name or his full name. The cultural meaning of an Icelander's last name is not that it is a part of one's name, but a short description of who one is. Halldór is Ásgrímsson — a son of Ásgrímur. Legally, it is a part of his name. Culturally, it is a definition of who begat whom, even if that definition is seemingly vague.
This also means that if there are for example two men named Jón in the same group, one named Jón Stefánsson and the other Jón Þorlásson, one could address Jón Stefánsson ´Jón Stefáns and Jón Þorláksson Jón Þorláks. When someone was talking to them he would not have to use the ending -son. Then the father's name would be used like a nickname. This is not very common now but was often done in the past.
Another good example of the formal mode of address is the Icelandic singer and actress Björk. Björk is commonly mistaken for an artist's stage name, as with fellow musicians Sting and Bono. However, Björk is simply Björk Guðmundsdóttir's given name, as any Icelander would address her, whether formally or casually.
As a result of using patronymics, in a four-person family there might be four different last names: the married couple Jón Stefánsson and Bryndís Atladóttir, and their children, Fjalar Jónsson and Katrín Jónsdóttir. With metronymics, the children would be Fjalar Bryndísarson and Katrín Bryndísardóttir.
This occasionally causes problems for Icelanders traveling abroad, especially with young children, as customs personnel suspect kidnapping of children.