Subsonic and low-supersonic propulsion systems based on detonation waves have the potential to substantially improve efficiency and power density compared to traditional engines. Numerous technical challenges remain to be solved in such systems, however, including obtaining more efficient injection and mixing of air and fuels, more reliable detonation initiation, and better understanding of the flow leaving the detonation chamber. These challenges can be addressed using numerical simulations. Such simulations are enormously challenging, however, since accurate descriptions of highly unsteady flow fields are required in the presence of combustion, shock waves, fluid-structure interactions, and other complex physical processes. In this talk, we present high-resolution two- and three- dimensional large eddy simulations of pulsed and rotating detonation engines and examine unsteady and turbulent flow effects on the operation, performance, and efficiency of the engine. These simulations are further used to test the accuracy of common Reynolds averaged turbulence models.