The maximum principle is one of the most useful and best known tools employed in the study of partial differential equations. The maximum principle enables us to obtain information about uniqueness, approximation, boundedness and symmetry of the solution, bounds for the first eigenvalue, quantities of physical interest, necessary conditions of solvability for some boundary value problems, etc. The book is divided into two parts. Part I contains two chapters and presents the classical maximum principle for linear equations, some of its direct extensions for nonlinear equations and their applications. Part II of this book is divided into three chapters and is devoted to the P function method and its applications. The book is addressed to graduate students in mathematics and to professional mathematicians, with an interest in elliptic partial differential equations.