Issues on the transport of one dimensional systems

In this paper, we will critically discuss recent theoretical and experimental developments on the transport of one dimensional quantum systems. In particular, we will focus on open issues and controversial results related to the finite temperature dynamics of integrable quantum models. These singular systems are commonly used in the description of quasi-one dimensional materials. They have recently attracted a lot of interest in connection to the interpretation of experiments, specially after the discovery of unusual thermal conductivity in magnetic compounds.