  In this Colloquium recent advances in the field of quantum heat transport are
reviewed. This topic has been investigated theoretically for several decades,
but only during the past twenty years have experiments on various mesoscopic
systems become feasible. A summary of the theoretical basis for describing heat
transport in one-dimensional channels is first provided. Then the main
experimental investigations of quantized heat conductance due to phonons,
photons, electrons, and anyons in such channels are presented. These
experiments are important for understanding the fundamental processes that
underly the concept of a heat conductance quantum for a single channel. Then an
illustration on how one can control the quantum heat transport by means of
electric and magnetic fields, and how such tunable heat currents can be useful
in devices is given. This lays the basis for realizing various thermal device
components such as quantum heat valves, rectifiers, heat engines,
refrigerators, and calorimeters. Also of interest are fluctuations of quantum
heat currents, both for fundamental reasons and for optimizing the most
sensitive thermal detectors; at the end of the review the status of research on
this intriguing topic is given.
