In Iceland the use of surnames is forbidden by a law, which passed in 1925.
There are a few exceptions: If a family had a surname before 1925, they have been allowed to keep using it.
Today many people in Iceland would like to take a hereditary surname modelled on the Norwegian, Swedish and Danish pattern and a change of the law might be possible.
Instead of surnames the Icelanders usually use primary patronymics according to the Old Norse tradition.
To create an Icelandic primary patronym, the suffix -son (= 'son') or -dóttir (= 'daughter') is added to the genitive form of the father's name. You will find the genitive forms of Icelandic names in the grammar tables on the respective name pages.
A new neutral suffix was introduced in Iceland in 2019. Since then, persons who are officially registered with non-binary gender are permitted to use the suffix -bur ('offspring') instead of -son or -dóttir.