Abstract <p> We survey results on the structural stability of shock waves in elastodynamics of compressible neo-Hookean materials. By nonlinear structural stability of a shock wave we mean the local-in-time existence and uniqueness of the discontinuous shock front solution of the elastodynamics equations, which guarantees the real existence of the shock wave as a physical structure. We describe finding structural stability conditions for shock waves in 2D elastodynamics using both the energy method and spectral analysis of the corresponding linearized free boundary problem. We also briefly discuss recent results on structural stability in the general 3D case. </p>