The continuum, inviscid‐flow equations of gas dynamics are used to predict the effect of thermal radiation on the internal structure of a shock wave. Numerical solutions of the governing equations show that considerable variation in the nature of the temperature and velocity profiles occurs, depending on the magnitude of the over‐all velocity change and the relative strengths of the radiative and convective energy fluxes. A unique feature of the work is the demonstration that, for a range of parametric values, discontinuities necessarily arise in the temperature and velocity profiles. The results include: (a) numerical integration of the basic equations for a representative range of parameters; (b) an analytic study of the equations by means of expansion procedures; (c) a study of the uniqueness of the solutions; (d) proof that previously published investigations cannot be generally applicable since they are restricted to continuous solutions; (e) prediction within a strong shock of a temperature maximum considerably larger than that corresponding to the Rankine‐Hugoniot conditions.