Conservation laws provide partial differential equations governing the flow of conserved quantities. These equations are fundamental and ubiquitous in physical theories from continuum mechanics. Classical examples include gas flow and elasticity, but also models for traffic flow, dynamics of crowds, granular flow etc. are covered by this class of equations.

The talk will focus on inviscid equations where dissipative effects areignored. We shall consider the Cauchy problem for systems of such equationsin one and several space dimensions. To highlight the issues particular to this class of equations (loss of regularity, on-uniqueness, entropy conditions...)we discuss some recent results (both positive and negative).