This book provides a comprehensive introduction to the mathematical theory of compressible flow, describing both inviscid and viscous compressible flow, which are governed by the Euler and Navier-Stokes equations respectively. The method of presentation allows readers with different backgrounds to focus on various modules of the material, either in part or more fully. Chapters include detailed heuristic arguments providing motivation for technical details that are rigorously presented later on in the text. This especially concerns the existence theory for steady and unsteady Navier-Stokes equations of isentropic compressible flow, and two by two systems of Euler equations in one space dimension. These parts are presented in a textbook style with auxiliary material in supporting sections and appendices. The book includes a rich index and an extensive bibliography, which allow quick orientation on the book itself as well as in the vast literature about the mathematical theory of compressible flow.