The aim of the study is to prove the existence of functions satisfying a quasilinear elliptic equation of general form and satisfying the given boundary conditions in Sobolev spaces. Methods of functional analysis, theory of functions of many variables, differential calculus of functions were used to achieve this goal. The main results of the study include the formulation and proof of the theorem on the existence of solutions of boundary value problems for quasilinear elliptic equations of general form in Sobolev spaces. A general methodology has been developed to prove the existence of solutions for a wide class of quasilinear elliptic equations.