<p>We demonstrate unprecedented control and enhancement of thermal radiation using subwavelength conical membranes of silicon nitride. Based on fluctuational electrodynamics, we find that the focusing of surface phonon-polaritons along these membranes enhances their far-field thermal conductance by three orders of magnitude over the blackbody limit. Our calculations reveal a non-monotonic dependence of the thermal conductance on membrane geometry, with a characteristic radiation plateau emerging at small front widths due to competing effects of the polariton focusing and radiative area. The obtained results thus introduce the conical geometry as a powerful degree of freedom for tailoring thermal radiation, with potential implications for energy harvesting and thermal management at the nanoscale.</p>