This book is a self-contained exposition of the theory of second order quasilinear elliptic partial differential equations, with emphasis on the Dirichlet problem in bounded domains. It is based on lecture notes from graduate courses at Stanford University, and extends beyond the scope of these courses. The book includes preparatory chapters on potential theory and functional analysis to make it accessible to a broad audience. The authors aim to convey the many ingenious techniques developed in the study of elliptic equations. The book is divided into two parts: Part I covers linear equations, including Laplace's equation, Poisson's equation, and the Newtonian potential. Part II deals with quasilinear equations, including maximum and comparison principles, fixed point theorems, and equations in two variables. The book also includes chapters on Banach and Hilbert spaces, Sobolev spaces, generalized solutions, and regularity. The book provides a comprehensive treatment of elliptic partial differential equations, covering topics such as the weak maximum principle, the Dirichlet problem, Hölder estimates, and boundary estimates. It also includes a chapter on equations of mean curvature type and an appendix on boundary curvatures and the distance function. The book concludes with a bibliography and indexes.This book is a self-contained exposition of the theory of second order quasilinear elliptic partial differential equations, with emphasis on the Dirichlet problem in bounded domains. It is based on lecture notes from graduate courses at Stanford University, and extends beyond the scope of these courses. The book includes preparatory chapters on potential theory and functional analysis to make it accessible to a broad audience. The authors aim to convey the many ingenious techniques developed in the study of elliptic equations. The book is divided into two parts: Part I covers linear equations, including Laplace's equation, Poisson's equation, and the Newtonian potential. Part II deals with quasilinear equations, including maximum and comparison principles, fixed point theorems, and equations in two variables. The book also includes chapters on Banach and Hilbert spaces, Sobolev spaces, generalized solutions, and regularity. The book provides a comprehensive treatment of elliptic partial differential equations, covering topics such as the weak maximum principle, the Dirichlet problem, Hölder estimates, and boundary estimates. It also includes a chapter on equations of mean curvature type and an appendix on boundary curvatures and the distance function. The book concludes with a bibliography and indexes.