Fjölnir Stefánsson began his music studies in 1943 on the cello with Dr. Heinz Edelstein at the Reykjavík College of Music. Later he majored in theory and compostion studying with Jón Thorarinsson and graduated in 1954. Stefánsson continued his studies in London with Matyas Seiber from 1954 to 1958. After finishing his studies he returned to Iceland. He taught at the Reykjavík College of Music from 1958 to 1968 when he became a principal of the Kópavogur College of Music, a position he held until 1. September 2000 when he retired. Stefánsson was one of the founders of Musica Nova in 1959 and served ont the board of the Iceland Music Information Centre from its establishment in 1968 to 1984 and as a chairman from 1983-1984. He served on the board of the Performing Right Society (STEF) from 1974-1988. In 1994 Stefánsson was honorary artist of Kópavogur.