We discuss results on the structural stability of shock waves in compressible inviscid isentropic elastic flows. By nonlinear structural stability of a shock wave we mean the local-in-time existence and uniqueness of the discontinuous shock front solution to the system of compressible elastodynamics. We show that planar shock waves in elastodynamics are always at least weakly stable, and we find a condition necessary and sufficient for their uniform stability (in the sense of the fulfilment of the uniform Kreiss-Lopatinski condition for the linearized problem). Since the system of elastodynamics satisfies the Agranovich-Majda-Osher block structure condition, uniform stability implies structural stability of corresponding nonplanar shock waves. We describe finding 2D and 3D uniform/structural stability conditions by both the energy method and the spectral analysis. We show that, as in isentropic gas dynamics, all compressive shock waves are uniformly stable for convex equations of state. We also briefly discuss principal difficulties of the stability analysis for shock waves in nonisentropic elastodynamics (thermoelasticity).