Abstract: The article contains a review of the most important results obtained in the framework of the St. Petersburg State University Research School on Nonlinear PDEs (the O.A. Ladyzhenskaya–N.N. Uraltseva School). The main attention is paid to the works carried out at the university over the past 50 years. The first part of the review concerns the solvability and qualitative properties of solutions to boundary value problems for the second order scalar quasilinear elliptic and parabolic equations, as well as variational problems. The planned second part of the review will include sections on fully nonlinear equations and systems of equations, and on free boundary problems.