This textbook gives a comprehensive and practical treatment of all existing Riemann solvers for compressible fluid dynamics and their use in the upwind method of Godunov and its high-order extensions. Related upwind methods of the Flux Vector Splitting type are also included, as are modern centred TVD methods. Methodologies for solving realistic problems in one, two and three space dimensions for both Cartesian and non-Cartesian geometries are presented in detail. Additional information is provided on further developments of the techniques and possible applications to practical problems in a variety of disciplines. A list of over 400 relevant references is given. The book is most useful for post-graduate students in Applied Mathematics, Engineering, Physics, Computing and other scientific disciplines such as Meteorology, Oceanography, Hydraulics and Chemistry, for example. It can be used as a means for self-study by academics and computational practioners in indusstry and research laboratories or as a teaching aid for postgraduate and final-year undergraduate courses on numerical methods etc.