Contrary to most other places in the world, little ol' Iceland is not very fond of "surnames" and EVERYBODY is on a first name basis.
(Yes, even with the president! :)

This is because there are hardly any "family names" in Iceland...
Now you say "Haa?!?!  No family names?!?!  How do you recognice each other and who belongs to whom?!?!" 

Well, it's simple really.  The Icelandic people are descendants from Vikings and they kept track of people by recognizing who was their father, as in "Helgi son of Ólafur the Brave, son of Leifur the Lucky" (not an actual example!).  Thus was born the tradition of naming children after their father's first name.

So the way it works in Iceland is that boys are given first names and then their father's first name + "son" as a "surname".  Girls on the other hand are given first names and then their father's first name + "dóttir" (as in "daughter") as a "surname".

Let me give you an hypothetical example:
Father = Ragnar Helgason
Mother = Inga Bjarnadóttir
Their son = Leifur Ragnarsson
Their daughter = Anna Ragnarsdóttir

Referring to the family by their "surnames" is therefore pointless and everyone goes by their first names! :)  Women also don't take their husband's surname when they get married as it would mess everything up!

It's also worth mentioning that most Icelanders can trace back their family-line to the 13th century or so (Iceland was first settled in the 9th century), so this system appears to be not completely useless! :)

Still in the dark?  Mail me with your questions!