Icelandic names: children have different last names to their parents

It's also fascinatingly controlling that any new first names (first names that have not previously been used in Iceland) must be approved by the Icelandic Naming Committee: https://en.wikipedia.org/wiki/Icelandic_Naming_Committee

Also, a first name must be "gender-appropriate" (that is, you are not legally allowed to name a girl with a boy's name, or vice-versa).