 In 1928 Stepanov became a professor at Moscow State University and then in 1939 also the Director of the Institute of Mechanics and Mathematics, where he continued until his death in 1950.In two publications (1923 and 1925) Stepanov gave necessary and sufficient conditions for a function of two variables, defined on a set S of measure greater than zero, to have a total differential almost everywhere on S. He also worked on dynamical systems (extending the work of George Birkhoff), the qualitative theory of ordinary differential equations, and almost periodic functions (extending the work of Harald Bohr).In the qualitative theory of ordinary differential equations Stepanov wrote a well-known textbook with his student Viktor Nemytskii.