Hodge attended Edinburgh University, graduating M.A. in 1923. He then came to Cambridge and took the Mathematical Tripos. In 1926 he took up a teaching position at the University of Bristol, before returning to Cambridge in 1930 as a Research Fellow at St. John's College. He became the Lowndean Professor of Astronomy and Geometry at Cambridge in 1936. He held the position until 1970. In 1958 he was elected Master of Pembroke College. He retired as Master in 1970. His other honours included the Adams Prize in 1937 and the Copley Medal of the Royal Society in 1974. He was vice-president of the Royal Society from 1959 to 1965 and was knighted in 1959.