Understanding turbulence and mix in compressible flows is of fundamental importance to real-world applications such as chemical combustion and supernova evolution. The ability to run in three dimensions and at very high resolution is required for the simulation to accurately represent the interaction of the various length scales, and consequently, the reactivity of the intermixin species. Toward this end, we have carried out a very high resolution (over 8 billion zones) 3-D simulation of the Richtmyer-Meshkov instability and turbulent mixing on the IBM Sustained Stewardship TeraOp (SST) system, developed under the auspices of the Department of Energy (DOE) Accelerated Strategic Computing Initiative (ASCI) and located at Lawrence Livermore National Laboratory. We have also undertaken an even higher resolution proof-of-principle calculation (over 24 billion zones) on 5832 processors of the IBM system, which executed for over an hour at a sustained rate of 1.05 Tflop/s, as well as a short calculation with a modified algorithm that achieved a sustained rate of 1.18Tflop/s. The full production scientific simulation, using a further modified algorithm, ran for 27,000 timesteps in slightly over a week of wall time using 3840 processors of the IBM system, clockin a sustained throughput of roughly 0.6 teraflop per second (32-bit arithmetic). Nearly 300,000 graphics files comprising over three terabytes of data were produced and post-processed. The capability of running in 3-D at high resolution enabled us to get a more accurate and detailed picture of the fluid-flow structure - in particular, to simulate the development of fine scale structures from the interactions of long-and short-wavelength phenomena, to elucidate differences between two-dimensional and three-dimensional turbulence, to explore a conjecture regarding the transition from unstable flow to fully developed turbulence with increasing Reynolds number, and to ascertain convergence of the computed solution with respect to mesh resolution.