Methane-air diffusion flames near surfaces are modeled using numerical bifurcation theory. Ignition, extinction, and stability boundaries are studied as functions of fuel how rate and oxidizer strain rate, using three reaction mechanisms of varying complexity (a one-step reaction, a C-1 reaction mechanism, and the Gas Research Institute [GRI 1.2] mechanism). It is found that as the fuel flow rate increases, the ignition temperature increases, whereas the extinction temperature decreases. For sufficiently high fuel flow rates and low oxidizer strain rates, the flame separates from the surface, and extinction cannot be caused by thermal quenching. In the absence of surface heat loss, two distinct extinction branches are found, namely, a blowoff or stretch limit branch at high fuel flow rates and a thermal quenching limit branch at low fuel flow rates. The complexity of the reaction mechanism affects primarily the blowoff branch at high fuel flow rates but not the thermal quenching branch due to the O-2 leaking through the reaction zone at low strain rates. Surface radiation alters the bifurcation behavior, beyond an absolute stability curve. The theoretically predicted flame structure and stability limits compare well with experimental results. Implications for modeling of extinction of complex condensed fuels are also discussed.