Premixed CH4/O-2/CO2 flames were studied experimentally and numerically in a swirl-stabilized combustor. The bulk throat velocity of the incoming combustible mixture was maintained constant at 5.2 m/s. The oxygen fraction of the O-2/CO2 oxidizer was kept constant at 60 vol %, while the effect of the equivalence ratio was examined from the blowout limit to the flashback one. The LES computational model was successfully validated through comparisons with experimental data in terms of axial and radial temperature profiles, as well as predicted OH* concentration maps versus visual flame appearance. Three different flame structures were observed while varying the equivalence ratio, namely, (I) double conical flame, (II) corner stabilized flame, and (III) swirl-stabilized (V-shaped) flame. The double conical flame is observed at low equivalence ratios near the blowout limit. This flame stabilizes within both the inner and outer shear layers and has two distinct reaction zones with two inner recirculation zones (IRZs) separated by a colder, reaction-free corner recirculation zone (CRZ). The corner-stabilized flame is observed at higher equivalence ratios. The downstream reaction zone diminishes and merges with the upstream one to form a flame that stabilizes within a reacting CRZ. Increasing the equivalence ratio further induces a transition to the typical V-shape of swirl-stabilized flames with significantly stronger IRZ and weaker CRZ. This flame is thus stabilized by its IRZ. The V-shape prevails until flashback occurs when the hot reaction zone moves closest to the burner throat.