  We review the progress that has been recently made in the application of
time-dependent density functional theory to thermoelectric phenomena. As the
field is very young, we emphasize open problems and fundamental issues. We
begin by introducing the formal structure of \emph{thermal density functional
theory}, a density functional theory with two basic variables -- the density
and the energy density -- and two conjugate fields -- the ordinary scalar
potential and Luttinger's thermomechanical potential. The static version of
this theory is contrasted with the familiar finite-temperature density
functional theory, in which only the density is a variable. We then proceed to
constructing the full time-dependent non equilibrium theory, including the
practically important Kohn-Sham equations that go with it. The theory is shown
to recover standard results of the Landauer theory for thermal transport in the
steady state, while showing greater flexibility by allowing a description of
fast thermal response, temperature oscillations and related phenomena. Several
results are presented here for the first time, i.e., the proof of invertibility
of the thermal response function in the linear regime, the full expression of
the thermal currents in the presence of Luttinger's thermomechanical potential,
an explicit prescription for the evaluation of the Kohn-Sham potentials in the
adiabatic local density approximation, a detailed discussion of the leading
dissipative corrections to the adiabatic local density approximation and the
thermal corrections to the resistivity that follow from it.
