  In this review paper we aim at illustrating recent achievements in anomalous
heat diffusion, while highlighting open problems and research perspectives. We
briefly recall the main features of the phenomenon for low-dimensional
classical anharmonic chains and outline some recent developments on perturbed
integrable systems, and on the effect of long-range forces and magnetic fields.
Some selected applications to heat transfer in material science at the
nanoscale are described. In the second part, we discuss of the role of
anomalous conduction on coupled transport and describe how systems with
anomalous (thermal) diffusion allow a much better power-efficiency trade-off
for the conversion of thermal to particle current.
