Fuglede graduated from Skt. Jørgens Gymnasium 1943 and received his mag. scient. og cand. mag. in 1948 at the University of Copenhagen after which he studied in USA until 1951. In 1952 he was employed as scientific assistant at Den Polytekniske Læreanstalt and in 1954 as amanuensis at Matematisk Institut University of Copenhagen, in 1958 associate professor, and in 1959 head of department. Fuglede also spent one year in Lund (Sweden) as Nordic docent. Fuglede received his dr.phil. (Ph.D.) in 1960 from the University of Copenhagen; his doctoral advisor was Børge Jessen. after which he became professor of math at Danmarks tekniske Højskole. In 1965 he became professor of math at the University of Copenhagen, where he stayed until his retirement in 1992.