Stability of shock waves

The progress already made in studies of the stability of shock waves and some tasks for the future are reviewed. The following aspects of the problem are discussed: 1) stability of shock waves as hydrodynamic discontinuities irrespective of the events which occur in the relaxation zone of a wave (this aspect includes also the problem of stability of a shock wave sustained by a piston and anomalies in the reaction of a wave to external perturbation, including reflection and refraction of perturbations); 2) stability of flow in the relaxation zone (structural stability) of a shock wave. The stress is on the theoretical side of the problem. However, potential practical realization of the shock-wave instability criteria are also considered. Schemes of decay of unstable shock-wave discontinuities are discussed.