In this paper we give a brief survey on the problem of shock structure solutions in fluid dynamics. First the classical results of Gilbarg and Paolucci for the Fourier-Navier-Stokes fluids are considered and then we concentrate our attention to the Extended Thermodynamics models furnishing a set of hyperbolic systems. In the case of 13 moments Grad pointed out the existence of a critical Mach number for which theC1solutions breakdown. We discuss the problem of non existence ofC1solutions for a generic hyperbolic system not only for boundary conditions corresponding to Lax shocks but also for the case of a characteristic shock.