A numerical approach for the simulation of low speed compressible reacting flows will be presented. This approach is based on mixed high-order integration in time and spectral-type methods in space and has been verified using combined numerical and asymptotic techniques. Results from the application of this approach to the simulation of non-premixed opposed-jet configurations with detailed H2-O2 kinetics, including vortex-flame interactions, will be presented. Preliminary results from a combined linear stability analysis and DNS study in a non-premixed reacting mixing layer will also be discussed. These results, obtained using finite rate chemistry and unity Lewis number, indicate the existence of instability modes other than shear layer ones, for conditions close to extinction.