  It has been observed in many numerical simulations, experiments and from
various theoretical treatments that heat transport in one-dimensional systems
of interacting particles cannot be described by the phenomenological Fourier's
law. The picture that has emerged from studies over the last few years is that
Fourier's law gets replaced by a spatially non-local linear equation wherein
the current at a point gets contributions from the temperature gradients in
other parts of the system. Correspondingly the usual heat diffusion equation
gets replaced by a non-local fractional-type diffusion equation. In this
review, we describe the various theoretical approaches which lead to this
framework and also discuss recent progress on this problem.
