A highly rarefied gas, initially in equilibrium, is compressed by an infinite plane piston. The resulting flow is considered when the gas is unbounded and also when the gas is bounded by a second stationary wall. In the first case, the density ahead of the piston is analysed for specularly and diffusely reflecting surfaces. In the second case, the average temperature of the gas is found as a function of the wall separation for specularly reflecting surfaces. This (non-equilibrium) temperature is compared with that generated in the corresponding continuum flow which involves multiple shock wave reflexion. It is shown that, for a given average density gradient, the free molecule temperatures are very much higher than the continuum temperatures.