This book is an introduction to partial differential equations (PDEs) and the relevant functional analysis tools which PDEs require. This material is intended for second year graduate students of mathematics and is based on a course taught at Michigan State University for a number of years. The purpose of the course, and of the book, is to give students a rapid and solid research-oriented foundation in areas of PDEs, like semilinear parabolic equations, that include studies of the stability of fluid flows and, more generally, of the dynamics generated by dissipative systems, numerical PDEs, elliptic and hyperbolic PDEs, and quantum mechanics. In other words, the book gives a complete introduction to and also covers significant portions of the material presented in such classics as Partial Differential Equations by Avner Friedman, Geometric Theory of Semilinear Parabolic Equations by Dan Henry, and Semigroups of Linear Operators and Applications to PDEs by A. Pazy.