The thesis addresses the influence of compressible effects on the stability of deflagration waves. Due to the quasi-isobaric nature of slow flames, compressible effects in laminar flames are usually neglected. Nevertheless, turbulent deflagrations may propagate at substantially higher speeds, suggesting that compressible effects may play a role in their dynamics. In the present thesis, the stability of diffusion-dominated high-speed deflagrations is addressed. The deflagration is studied in the thickened regime, hence addressing the long wavelength limit of these deflagrations. The deflagrations are modelled by the compressible reactive Navier-Stokes equations with a single-step Arrhenius reaction model. The 2D stability of the steady traveling-wave solution is studied by direct simulation. It is found that, as the flame compressibility becomes significant, not only does the growth rates of the cellular profile of the deflagration waves increase, but the traditional correlation of the burning velocity and the flame surface area become far less strong. Significant compression regions form in front of the nonlinear flames. These compression regions have been found to increase the growth rates by increasing the temperature of the unburned gas in front of the flames, as well as convecting the flame forward. The results show that the flame propagation velocity in references to the unburned gas was significantly faster than the burning velocity. The vorticity was given consideration, as the compressibility of flame increase one can expect the baroclinic source to be of greater significance. The vorticity was show to, in effect, increase as compressibility increases while unexpectedly having a stabilizing direction of rotation on the cellular structure of the flames.