Sobolev introduced notions which are now fundamental in several different areas of mathematics. Sobolev spaces can be defined by growth conditions on Fourier transforms; they and their embedding theorems are an important subject in functional analysis. Generalized functions, later known as distributions), were introduced by Sobolev in 1935 and further developed by Laurent Schwartz; they redefined the notion of differentiation.

Sobolev graduated from the Leningrad University in 1929. He worked at the Steklov Mathematical Institute in Leningrad from 1932, and in Moscow from 1934. He headed the institute when it was evacuated to Kazan during the Second World War. He was a Moscow University professor from 1935 to 1957 and also a deputy director of the Institute for Atomic Energy 1943-57.

In 1956 Sobolev joined a number of prominent scientists in proposing a large-scale scientific and educational initiative for the Eastern parts of the Soviet Union, which resulted in the creation of the Siberian Branch of the Academy of Sciences. He was the founder and first director of the Institute of Mathematics in Novosibirsk, which was later to bear his name, and played an important role in the establishment and development of Novosibirsk State University.