The manner in which the Navier–Stokes equations of fluid mechanics break down under conditions of low-density, hypersonic flow is investigated numerically. This is performed through careful and detailed comparisons of solutions obtained with continuum and Monte Carlo simulation techniques. The objective of the study is to predict conditions under which the continuum approach may be expected to fail. Both normal shock waves and bow shocks formed by flow over a sphere are considered for argon and nitrogen. It is found that a Knudsen number based on local flow conditions and gradients is a convenient and accurate criterion for indicating breakdown of the continuum flow equations. Failure of the Navier–Stokes equations in hypersonic transitional flows occurs both in the shock front and in the region immediately adjacent to the body surface.