In this paper, the solution of classical and generalized linear and nonlinear elliptic partial differential equations of the second order with non-local boundary conditions is studied. The basic concepts and theorems related to the study are included, which includes the main result of this research. Many basic definitions, theorems and observations about Sobolev spaces H^k (Ω) and H_0^k (Ω) are discus-sed. The existence and uniqueness of the solution of the Poisson equation with non-local boundary conditions are considered. The existence and uniqueness of the solution of a second-order quasi-linear elliptic differential equation with non-linear integral boun-dary condition are also discussed. The argument for proving the previous problem is based on Banach's fixed point theorem in the complete metric space, the maxima and minima principle, and the comparison principle................ Key word:................. Sobolev Spaces, Poisson Equation, Quesilinear Ellipitic Differential Equation, Boundary, and Banach's fixed Point Theorem.