The theme of this book is the investigation of globally asymptotic solutions of evolutionary equations. Locally asymptotic solutions of the Navier - Stokes equations and reaction-diffusion equations are the starting point, and by considering perturbed evolutionary equations, global approximations are constructed. The lectures upon which this book is based were warmly received at the universities of Rome and Pavia, and at the Scuola Normale Superiore in Pisa. Here Professor Vishik has collated his lecture notes, and has added an appendix describing his recent work on attractors deriving from dynamical systems. This is unquestionably a fine addition to the Lezioni Lincee, and will be a necessary addition to the library of all who seek an insight into the solution of evolutionary equations.