Current understanding on the fundamental physico-chemical mechanisms governing the structure and stabilization of premixed and diffusion flames in subsonic and supersonic laminar and turbulent flows are classified and discussed, with emphasis on possible applications in supersonic propulsion. Specific topics discussed include the design concepts of supersonic engine operation and fuel injection, the ignition of combustibles in homogeneous and diffusive media, the extinction of premixed and diffusion flames through reactant leakage, heat loss, and aerodynamic stretching, the stabilization and liftoff of inverted, burner-stabilized, and rim-stabilized flames, and the various proposed mechanisms for the stabilization and blowout of jet diffusion flames. The fundamental similarities and differences between the various critical phenomena are indicated. Research topics of potential importance to supersonic combustion are suggested.