This research focuses on the development of wall functions suitable for the prediction of high-speed compressible flows. Wall-functions avoid the need for prohibitively expensive fine near-wall meshes and low-Re models of turbulence which still involve a certain amount of approximation. The conventional log-law-based wall functions, however, have limitations in even incompressible cases, which are further compounded when applied to high-speed compressible flows. The objective of this study is to examine the performance of an advanced analytical wall-function treatment which has been successfully used in a range of incompressible flows and explore how compressibility effects could be accounted for in such approaches. The starting point was the implementation of the analytical wall function proposed by Craft et al (2002) in OpenFoam and its subsequent use for the prediction of the impinging shock interaction and compression corner cases up to a Mach number of 3. The wall pressure and skin friction results obtained by the original version result in improvements over those of the standard wall function (log-law based) and are close to those obtained by the low-Re number modelling for supersonic flows. However, an unphysical behaviour is encountered when applying it to higher Mach number cases. A compressible flow version of the analytical wall function is proposed which includes the following modifications: a)inclusion of thermal dissipation terms in the analytical equation for the energy variation over the near-wall cells, b) Variable molecular viscosity (due to temperature variations) over the viscous sub-layer, c) improved variation of the convection terms in the near-wall cell analytical equations. The resultant model has been applied to the above flows up to Mach numbers of 9 and comparisons drawn with experimental data and with predictions from the log-law based wall functions and from the Low-Re Launder and Sharma model. The present results are consistently closer to the data than those of other wall functions in some instances even better than those of the low-Re number. Improvements are especially noticeable in the prediction of the wall heat flux rates, where the log-law wall function generally predicts too low values in the shock interaction region, while the low-Re model, predicts too high heat transfer rates in the highest Mach number cases, as a result of overpredicting turbulence levels where extremely rapid near-wall temperature variations are found.