  We derive shape-independent limits to the spectral radiative heat-transfer
rate between two closely spaced bodies, generalizing the concept of a black
body to the case of near-field energy transfer. Through conservation of energy
and reciprocity, we show that each body of susceptibility $\chi$ can emit and
absorb radiation at enhanced rates bounded by $|\chi|^2 / \textrm{Im} \chi$,
optimally mediated by near-field photon transfer proportional to $1/d^2$ across
a separation distance $d$. Dipole--dipole and dipole--plate structures approach
restricted versions of the limit, but common large-area structures do not
exhibit the material enhancement factor and thus fall short of the general
limit. By contrast, we find that particle arrays interacting in an idealized
Born approximation (i.e., neglecting multiple scattering) exhibit both
enhancement factors, suggesting the possibility of orders-of-magnitude
improvement beyond previous designs and the potential for radiative heat
transfer to be comparable to conductive heat transfer through air at room
temperature, and significantly greater at higher temperatures.
