It is named after Árni Magnússon, a 17th–18th century collector of medieval Icelandic manuscripts.
The Minister of Education, Science and Culture appoints a five-member board for four-year terms.
Professor Guðrún Nordal took over as director of the institute on 1 March 2009, appointed for a period of five years.
In 1927–28 four manuscripts and some 700 charters and other legal documents were returned to the Icelandic National Archives.
Ten years later, after the transfer of manuscripts from Copenhagen had begun in earnest, the laws concerning the institute were changed and it was renamed Stofnun Árna Magnússonar á Íslandi (generally referred to as Árnastofnun).