This book is devoted to new and classical results of the theory of linear partial differential operators in Gevrey spaces. The microlocal approach is adopted, using pseudo-differential operators, wave front sets and Fourier integral operators. Basic results for Schwartz-distributions, c and analytic classes are also included, concerning hypoellipticity, solvability and propagation of singularities. Also included is a self-contained exposition of the calculus of the pseudo-differential operators of infinite order.