Modeling computations are performed to highlight the importance of cavities to the stabilization of methane flames in micro-structured combustion systems. The effects of different design factors on flame stability are investigated. A dimensionless number analysis is performed to better understand the heat transfer characteristics of the burner. The results indicate that the recirculating, hot combustion products continuously contact and ignite the incoming fuel-air mixture, thus anchoring the flame in the vicinity of the recirculation region. When a cavity is used, the flame is anchored because the cavity induces recirculation of hot combustion products within the combustor. Combustion is stabilized and flame stability is improved by recirculation of hot combustion products induced by the cavity structure. An additional effect of the cavity is to direct the incoming fuel-air mixture outwards toward the walls of the combustor, thus rapidly diffusing the mixture into the combustor, making effective use of the combustor's volume. At low flame temperatures, long combustor residence times are needed for combustion completion in the gas phase. Flame stabilization is dependent upon speed of the fuel-air mixture entering the combustion region where propagation of the flame is desired. A sufficiently low velocity must be retained in the region where the flame is desired in order to sustain the flame. A region of low velocity in which a flame can be sustained can be achieved by causing recirculation of a portion of the fuel-air mixture already burned thereby providing a source of ignition to the fuel-air mixture entering the combustion region. However, the fuel-air mixture flow pattern, including any recirculation, is critical to achieving flame stability. The inlet velocity and wall thermal conductivity are vital in determining the flame stability of the burner. Faster ignition and more efficient heat transfer can be achieved, but the design is challenging due to the loss of flame stability.