We present an overview of a portion of the compressible turbulence program associated with the ASC-sponsored Center for Simulation of Dynamic Response in Materials. Progress in several areas will be summarized. The first is the development of a hybrid WENO-centered-finite-difference method suitable for the large-eddy simulation (LES) of the strong-shock, Richtmyer-Meshkov (RM) instability in a rectangular tube, with end-wall shock reflection and reshock of the evolving mixing layer. The use of this scheme together with a structure-based SGS model for LES of RM instability will be outlined. Second, results obtained from DNS simulations of nonpremixed turbulent combustion in a three-dimensional jet will be described. These were done with finite-rate chemistry using the four-step reduced mechanism of Peters. The emphasis is on diagnostics relating to statistical analysis of flame geometry and turbulent mixing across the range of scales. Lastly, we present results obtained from DNS of Rayleigh-Taylor instability. They elucidate the physics of the gravity-driven mixing through analysis of PDFs of mole fraction and scalar dissipation, and show the structure of the scalar-dissipation field, amplified through the incorporation and extensions of level-set analysis.