Thykkvabaearklaustur is presently a church site in the county Alftaver, and was formerly the location of a monastery, which was occupied from its foundation in 1168 to the Reformation in 1550.

One of the monks of this monastery in the 14^th century, Eysteinn Asgrimsson, composed the poem „Lilja", one of Iceland’s most famous medieval devotional poems. A pillar of columnar basalt stands where the monastery is believed to have stood.