A calculation procedure for modelling the interaction between shock waves and attached or separated turbulent boundary layers is introduced. The numerical framework, applicable to general curved grids, combines cell-vertex storage, a Lax-Wendroff time-marching scheme and multigrid convergence acceleration. The main numerical ingredients of the procedure are documented in some detail, with particular emphasis placed on the inclusion of viscous and turbulence transport within the cell-vertex framework, which was originally formulated for inviscid flow. An investigation of the predictive performance of alternative transport models of turbulence has been the primary objective of the present work. Particular attention has been focused on a comparison between variants of low Reynolds number k-ε models and an algebraic variant of a Reynolds-stress transport closure in strong interaction situations, including shock-induced separation. The turbulence models are introduced, and important numerical issues affecting their stable implementation are discussed. The calculation procedure is then applied to two confined transonic flows over bumps — one incipiently and the other extensively separated (Delery Cases A and C) — and to the transonic flow around the RAE 2822 aerofoil at two angles of incidence (Cases 9 and 10). The investigation demonstrates that the eddy-viscosity models tend to seriously underestimate the strength of interaction, particularly when separation is extensive. The performance of the Reynolds-stress model is not entirely consistent across the range of conditions examined. In the case of bump flows, the model displays strong sensitivity to the shock, predicting excessive boundary layer displacement in Case A, a broadly correct separation pattern in Case C and insufficient rate of post shock recovery in both cases. The aerofoil flows are either attached or incipiently separated, and the benefits arising from Reynolds-stress modelling are modest. Neither the k-ε model nor the Reynolds-stress closure is able to return a satisfactory representation of the most challenging RAE 2822 Case 10; at least not with the recommended windtunnel corrections for freestream Mach number and angle of incidence.