These notes were developed for a graduate-level course on the theory and numerical solution of nonlinear hyperbolic systems of conservation laws. Pa rt 1 deals with the basic mathematical theory of the equations: the notion of weak solutions, entropy conditions, a detailed description of the wave structure of solutions to the Riemann problem. The emphasis is on tools and techniques that are indispensible in developing good numerical methods for discontinuous solutions. Part 2 is devoted to the development of high resolution shock-capturing methods, including the theory of total variation diminishing (TDV) methods and the use of limiter functions. This book is intended for a wide audience and should be of use both to numerical analysts and to computational researchers in a variety of applications.