This is a self-contained exposition of the well-posedness theory for nonlinear hyperbolic systems of conservation laws, completed by the author together with his collaborators. The systems of partial differential equations under consideration arise in many areas of continuum physics. No familiarity with the subject is assumed, so the book can be used by graduate students and researchers interested in developments concerning nonlinear partial differential equations and the mathematical aspects of shock waves and propagating phase boundaries. The text covers the existence, uniqueness and continuous dependence of classical (compressive) entropy solutions. It also introduces the reader to the developing theory of nonclassical (undercompressive) entropy solutions. The study of nonclassical shock waves is based on the concept of a kinetic relation introduced by the author for general hyperbolic systems and derived from singular limits of hyperbolic conservation laws with balanced diffusion and dispersion terms.