Hamburger obtained his Ph.D. from the University of Munich in 1914 under the supervision of Alfred Pringsheim and after war service obtained his Habilitation for a thesis on "Extensions of the Stieltjes moment problem". He was appointed Privatdozent at the University of Berlin in 1921 and professor at the University of Cologne in 1926. He left Cologne in 1935, after the imposition of the Nuremberg Laws, and returned to his mother's home in Berlin. In 1939, he left Germany, and from 1941 to 1946 he was lecturer at the University of Southampton. After the war, he received an invitation to return to Cologne, but instead moved to the University of Ankara in 1947. He returned to Cologne in 1953.