  We present a detailed derivation of heat radiation, heat transfer and
(Casimir) interactions for N arbitrary objects in the framework of
fluctuational electrodynamics in thermal non-equilibrium. The results can be
expressed as basis-independent trace formulae in terms of the scattering
operators of the individual objects. We prove that heat radiation of a single
object is positive, and that heat transfer (for two arbitrary passive objects)
is from the hotter to a colder body. The heat transferred is also symmetric,
exactly reversed if the two temperatures are exchanged. Introducing partial
wave-expansions, we transform the results for radiation, transfer and forces
into traces of matrices that can be evaluated in any basis, analogous to the
equilibrium Casimir force. The method is illustrated by (re)deriving the heat
radiation of a plate, a sphere and a cylinder. We analyze the radiation of a
sphere for different materials, emphasizing that a simplification often
employed for metallic nano-spheres is typically invalid. We derive asymptotic
formulae for heat transfer and non-equilibrium interactions for the cases of a
sphere in front a plate and for two spheres, extending previous results. As an
example, we show that a hot nano-sphere can levitate above a plate with the
repulsive non-equilibrium force overcoming gravity -- an effect that is not due
to radiation pressure.
