  We present a comprehensive study of heat transport through small
superconducting point contacts. The heat current for a temperature biased weak
link is computed as a function of temperature and barrier transparency of the
junction. The transport of thermal energy is controlled by the quasiparticle
transmission probability for the point contact that couples the superconducting
leads. We derive this transmission probability and results for the heat current
starting from nonequilibrium transport equations and interface boundary
conditions for the Keldysh propagators in quasiclassical approximation. We
discuss the thermal conductance for both clean and dirty superconducting leads,
as well as aspects of the nonlinear current response. We show that the
transmission probability for continuum quasiparticle states is both energy- and
phase-dependent, and controlled by an interface Andreev bound state below the
continuum. For high transparency barriers the formation of a low-energy bound
state, when the phase is tuned to $\phi=\pi$, leads to a reduction of the heat
current relative to that for $\phi=0$. For low-transparency barriers, a shallow
Andreev bound state just below the continuum edge is connected with resonant
transmission of quasiparticles for energies just above the gap edge, and leads
to enhanced heat conductance as the temperature is lowered below the
superconducting transition.
