It was recognized soon after the pioneering works of Riemann and Stokes in the mid-nineteenth century that entropy increases as the gas flows across a shock wave for polyatomic gases. Around 1940's Bethe and Weyl independently formulated the convexity condition for the equivalence of the compressibility of a shock and the entropy increase across it. This was subsequently generalized to the general system of hyperbolic conservation laws by Lax. The situation without convexity is interesting. The Russian school of Oleinik and Krushkov obtained complete results for scalar laws. It is understood now that the existence of entropy for a system is a constitutive hypothesis. Godunov established the relation between the existence of entropy and the symmetric structure of a system. There have been efforts to relate the admissibility conditions for shock waves to the entropy production. For this, we offer a definitive result for shock waves in the Euler equations for compressible media. In this talk, we will survey the historical developments on general systems as well as some exact analysis for the Euler equations.