This textbook provides a self-contained and elementary introduction to the modern theory of pseudodifferential operators and their application to partial differential equations. It presents the necessary material on Fourier transformation and distribution theory, the basic calculus of pseudodifferential operators on the n-dimensional Euclidean space, an introduction to the theory of singular integral operators, the modern theory of Besov and Bessel potential spaces, and several applications to wellposedness and regularity question for elliptic and parabolic equations. The basic notation of functional analysis needed in the book is introduced and summarized in the appendix.