Sobolev studied Mathematics at Leningrad University and graduated in 1929, having studied under Professor Nikolai Günther. After graduation, he worked with Vladimir Smirnov, whom he considered as his second teacher. He worked in Leningrad from 1932, and at the Steklov Mathematical Institute in Moscow from 1934. He headed the institute in evacuation to Kazan during World War II. He was a Moscow State University Professor of Mathematics from 1935 to 1957 and also a deputy director of the Institute for Atomic Energy from 1943 to 1957 where he participated in the A-bomb project of the USSR.