V.A. Steklov's work on equations of mathematical physics and development of his results in this field

This paper is an extended account of the author's talk at the International Conference “Contemporary Problems of Mathematics, Mechanics, and Mathematical Physics” dedicated to the 150th anniversary of Vladimir Andreevich Steklov. Steklov's main studies on the solvability of boundary value problems for equations of mathematical physics are briefly described, and the further development of this field of research is surveyed. The main attention is focused on the statements of the Dirichlet problem and the conditions on the domain and given functions under which solvability theorems are valid.