The primary objective of the research is to use Direct Numerical Simulations (DNS) to study turbulent non-premixed combustion. In DNS, the fluid mechanical and thermochemical conservation equations are solved by an accurate numerical method, without any averaging or turbulence modeling. In principle, then, DNS could be used to study a turbulent diffusion flame, for example. In practice, however, computational limitations severely restrict the flows that can be simulated. For non-reacting flows, DNS is restricted to simple geometries and moderate Reynolds number. For reacting flows there are severe restrictions on the thermochemistry. Our approach is to use DNS to study very simple turbulent reactive flows, that contain qualitatively the same phenomena real flames. Based on the insights and information gained, statistical models will be developed and tested. These models are then applicable to the turbulent flames of practical importance.