  We provide a brief and comprehensive overview over recent developments in the
field of phase-coherent caloritronics in ordinary and topological Josephson
junctions. We start from the simple case of a short, one-dimensional
superconductor-normal metal-superconductor (S-N-S) Josephson junction and
derive the phase-dependent thermal conductance within the Bogoliubov-de Gennes
formalism. Then, we review the key experimental breakthroughs that have
triggered the recent growing interest into phase-coherent heat transport. They
include the realization of thermal interferometers, diffractors, modulators and
routers based on superconducting tunnel junctions. Finally, we discuss very
recent theoretical findings based on superconductor-topological
insulator-superconductor (S-TI-S) Josephson junctions that show interesting
heat transport properties due to the interplay between topological band
structures and superconductivity.
