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It is well established that near-field radiative heat transfer (NFRHT) can exceed Planck’s blackbody limit1 by orders of magnitude owing to the tunneling of evanescent electromagnetic frustrated and surface modes2-4, as has been demonstrated experimentally for NFRHT between two large parallel surfaces5-7 and between two subwavelength membranes8,9. However, while nanostructures can also sustain a much richer variety of localized electromagnetic modes at their corners and edges,10,11 the contributions of such additional modes to further enhancing NFRHT remain unexplored. Here, we demonstrate both theoretically and experimentally a new physical mechanism of NFRHT mediated by these corner and edge modes, and show it can dominate the NFRHT in the “dual nanoscale regime” in which both the thickness of the emitter and receiver, and their gap spacing, are much smaller than the thermal photon wavelengths. For two coplanar 20 nm thick SiC membranes separated by a 100 nm vacuum gap, the NFRHT coefficient at room temperature is both predicted and measured to be 830 W/m2K, which is 5.5 times larger than that for two infinite SiC surfaces separated by the same gap, and 1400 times larger than the corresponding blackbody limit accounting for the geometric view factor between two coplanar membranes. This enhancement is dominated by the electromagnetic corner and edge modes which account for 81% of the NFRHT between the SiC membranes. These findings are important for future NFRHT applications in thermal management and energy conversion.