Any mathematical model of a continuum is given by a system of partial differential equations (PDEs). In continuum mechanics, the conservation laws of mass, momentum and energy form a common starting point, and each medium is then characterized by its constitutive laws. The conservation laws and constitutive equations for the field variables, under quite natural assumptions, reduce to field equations, i.e., partial differential equations, which, in general, are nonlinear and nonhomogeneous. For nonlinear problems, neither the methods of their solutions nor the main characteristics of the motion are as well understood as in the linear theory. Before we proceed to discuss mathematical concepts and techniques to understand the phenomena from a theoretical standpoint and to solve the problems that arise, we introduce here the hyperbolic systems of conservation laws in general (see Benzoni-Gavage and Serre [13]), and then present some specific examples, for application or motivation, which are of universal interest.