An exact similarity solution of the compressible-flow Navier–Stokes equations is presented, which embeds supersonic, transonic, and subsonic regions. Describing the viscous and heat-conducting high-gradient flow in a shock wave, the solution accommodates non-linear temperature-dependent viscosity as well as heat-conduction coefficients and provides the variation of all the flow variables and their derivatives. Also presented are methods to obtain time-dependent and/or multi-dimensional solutions as well as verification benchmarks of increasing severity. Comparisons between the developed analytical solution and CFD solutions of the Navier–Stokes equations, with determination of convergence rates and orders of accuracy of these solutions, illustrate the utility of the developed exact solution for verification purposes. Copyright © 2012 John Wiley & Sons, Ltd.