The stability of methane/air and hydrogen/air flames in an axisymmetric counterflow burner was investigated experimentally for different burner geometries, degrees of fuel dilution, and combinations of flow velocities. Both planar diffusion flames and edge flames were observed, and the transitions between these flame types were studied. The experimental results confirmed previously published numerical predictions on diluted hydrogen/air flames: the existence of two distinct stable flame types; the possibility of switching between the two flame types by perturbing the flames, e.g., by suitably changing a flow velocity; and the strong hysteresis for the transition from one flame type to the other. Flame stability diagrams were compiled which delineate the range of fuel and air flow velocities for which the planar diffusion flame and the toroidal edge flame are stable. The lower boundary curve for the edge flame stability exhibits a characteristic minimum at a well-defined value of the fuel velocity. For fuel velocities lower than this value, the transition between the edge and the diffusion structure is reversible, and the flames exhibit bistable behavior. For higher fuel velocities, the decrease of air velocity leads to the extinction of the edge flame. An investigation of both the cold and the reactive flow field identified bistable behavior for the flow field as well. Except for very low flow rates, the stagnation plane stabilizes in two positions, close to either of the two nozzles. Detailed numerical simulations of hydrogen flames capture the essentials of this behavior. The observed flame extinction results from the interaction of the flame dynamics with the dynamics of the flow field.