Modern Icelanders still use the Old Norse patronymic system: a form of their father's given name ending in '-sson' or '-sonur' for a son and -dattir or -dottir for a daughter. The telephone book is arranged by first names and they are always used, even on formal occasions.
In some cases, a family name can be inherited (usually dating from the period of Danish rule) but it is not legal to invent a new one. Recently, the naming laws have been changed to allow the mother's name, or those of both parents, to be used. A woman does not take her husband's name on marriage. There was also a rule that only historically correct names could be used but this has been relaxed recently.