This volume is concerned with a few basic results in the classical and modern theory of boundary value problems for elliptic, parabolic and wave equations. The emphasis is on understanding the main results of the field, along with a few classical and modern methods. Weak solutions to boundary value problems of parabolic and hyperbolic type are also included. Furthermore, topics such as the maximum principle and potentials are presented. The book can be recommended for a graduate level course, and should be of interest to researchers and graduate students whose work involves partial differential equations, mathematics of physics and of engineering, potential theory and calculus of variations.