We survey recent achievements in anomalous heat diffusion in low-dimensional classical
anharmonic lattices. In one dimension, models with three conservation laws
belong tho the universality class of the famous Kardar-Parisi-Zhang equation.
We outline some recent developments on i) perturbed integrable systems on
ii) the effect of long-range forces. The topic is of interest
for the general theory of nonequilibrium steady states but also for nanoscale
heat transfer.