Books have been written in the country almost since the day it was first discovered, but oldest surviing manuscripts are from th e13th century or so, some of the are copies of older texts however.

Modern Literature

Icelandic literature and in particular the art of writing fiction lost much of it´s spark in the early 20th century and has not really recovered since, although the country is still the most literate country in the world and there are published more books per head than anywhere else. The most famous of modern icelandic authors is Halldór Kiljan Laxnes (1902-1998), he received the Stalin Price for literature in 1953 and the Nobel Price in 1955.

The Eddas

These two are not only the most important to have been written in Icelandic they are arguably also the most important books tho have been written in Europe during hte middle ages. They where written in the 12th century, while the subject matter is technically poetry and it´s construction, the poems and poetic language was so firmly rooted in germanic mythology as to make those 2 books the most valuable source material on mythology that we can find.
The Elder Edda is a collection of sacred and secular poetry and is sometimes refered to as the "Poetic Edda". It´s commonly believed that the majority of the proetry collected in the book dates from the 9th to the 12th century.
The the younger Edda is however a tretisise on the writing of poetry and the Ásatrú/Germanic Mythology that much of such poetry was based on, it was written by Snorri Sturluson in the 12th Century, it´s commonly refered to as the "Prose Edda".