In what country do people use patronymic names instead of traditional surnames?

Icelandic names are names used by people from Iceland. Icelandic surnames are different from most other naming systems in the modern Western world by being patronymic or occasionally matronymic: they indicate the father (or mother) of the child and not the historic family lineage. Iceland shares a common cultural heritage with the Scandinavian countries of Denmark, the Faroe Islands, Norway, and Sweden. Unlike other Nordics, Icelanders have continued to use their traditional name system, which was formerly used by all Nordic countries except partly Finland.

The Icelandic system is thus not based on family names (although some people do have family names and might use both systems). Generally, with few exceptions, a person's last name indicates the first name of their father (patronymic) or in some cases mother (matronymic) in the genitive, followed by -son ("son") or -dóttir ("daughter").

A man named Jón Einarsson has a son named Ólafur. Ólafur's last name will not be Einarsson like his father's; it will become Jónsson, indicating that Ólafur is the son of Jón (Jóns + son). The same practice is used for daughters. Jón Einarsson's daughter Sigríður's last name would not be Einarsson but Jónsdóttir. Again, the name means "Jón's daughter" (Jóns + dóttir).