The anharmonic properties of low-dimensional carbon crystal lattices are reviewed. The energy and crystal momentum conservation rules in two- and one-dimensional crystals lead to a drastic reduction of the phase space available for anharmonic phonon decay. This is illustrated with first principles calculations of the anharmonic properties of graphite and graphene. Experimental Raman linewidth data for the Radial Breathing Mode (RBM) in suspended single-walled carbon nanotubes are also interpreted in terms of a simple model in which a phonon decay bottleneck induced by the low dimensionality leads to a population time dependence in which a fast initial decay is followed by a slow decay determined by the decay rate of a large population of secondary phonons. These results are key to understanding the combined dynamics of electrons and phonons that determines the electrical transport properties in low-dimensional carbon nanostructures. In the case of the RBM in carbon nanotubes, they raise the intriguing possibility of using the linewidth of the Raman peak to determine the, chirality of the nanotube. (C) 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim