High order simulations are necessary in order to capture fine details in resolving supersonic reactive flows. However, high Mach number compressible flows feature sharp gradients and discontinuities, which present a challenge to successful simulations using high order methods. Spectral methods have proven a powerful tool in simulation of incompressible turbulent flows, and recent advances allow the application of spectral methods to compressible reactive flows. We review the recent advances in the theory and application of spectral methods which allow stable computations of discontinuous phenomena, and the recovery of high order information via postprocessing, and present applications of high Mach number reactive flows. To cite this article: D. Gottlieb, S. Gottlieb, C. R. Mecanique 333 (2005).