If you are applying for Icelandic citizenship, you can apply for your child that is younger than 18 in the same application. The application process is online.

If your child is between 18 and 20 years old it may be sufficient to send the Directorate of Immigration a notification that your child wishes to acquire Icelandic citizenship instead of an application.

Specific rules also apply to children or young persons who are stateless or have been granted international protection and for children of Icelandic citizens.

Unfortunately, most of the linked information is yet only available in Icelandic.