All entirely new approach to the large-eddy simulation (LES) of high speed compressible turbulent flows is presented. In this new approach to large-eddy simulations, subgrid scale stress models go continuously to Reynolds stress models in the coarse mesh/infinite Reynolds number limit, where a Reynolds stress calculation (RANS) is done in parallel with the LES to get an estimate of the Kolmogorov length scale. Hence, this formally constitutes a combined LES/time dependent RANS capability. Here, the Reynolds stress model that is recovered in the coarse mesh limit has a dependence on rotational strains through strain-dependent coefficients as well as through anisotropic eddy viscosity terms which are usually neglected in subgrid scale models. Unlike in existing subgrid scale models, this new model has the correct dependence on the mesh size through the dimensionless ratio of the computational mesh size to the Kolmogorov length scale. It is this ratio that determines how well-resolved a computation is in the numerical simulation of turbulence and should be used to parameterize subgid scale models. Use is made of the Morkovin hypothesis in the formulation of compressible subgrid scale models. This allows for the description of supersonic turbulent flows provided that they are not in the hypersonic flow regime. A critical assessment of this new approach is provided along with a discussion of traditional approaches and the prospects for future applications to the high-speed compressible flows of technological importance.