The talk will present a survey of recent results and current research in the theory of hyperbolic conservation laws. Recent analysis has shown that front tracking, vanishing viscosity and semidiscrete approximations preserve a uniform bound on the total variation of the solutions. All these approximations converge to a unique limit, depending continuously on the initial data in the L^1 norm. On the other hand, fully discrete numerical schemes can generate an arbitrary large amount of oscillations. For general hyperbolic systems, the convergence of these numerical schemes remains an open problem.