Graphyne, a two-dimensional planar carbon allotrope and consisting of sp and sp2 carbon atoms, receives great attention in a wide community of scientists and engineers beyond graphene. Herewith we investigate the thermal transport property of graphyne and graphyne nanoribbons by performing nonequilibrium molecular dynamics simulation. Our simulations reveal abnormal thermal transport in graphyne that exhibits a low thermal conductivity (as low as 8 W/mK at room temperature), which is two to three orders of magnitude lower as compared to graphene. Detailed lattice dynamic calculations and phonon polarization analysis suggest that, the intrinsically low thermal conductivity of graphyne originates from the localization and the domination of the low frequency in-plan longitudinal modes to the acetylenic linkages and the large lattice vibration mismatch between the linkages and the hexagonal rings, which induces inefficient energy transfer between the soft phonon modes in the linkages and the stiffer vibrational modes in the hexagonal rings. We also illustrated an intriguing width dependent thermal conductivity of graphyne nanoribbons, which is fundamentally different from that of graphene nanoribbons and stems from the surface dominated phonon modes presented in the graphyne edge. Our simulations provide a detailed physical picture of thermal transport in graphyne and could offer useful guidance for engineering the thermal transport properties of graphyne for applications of graphyne related devices such as thermoelectrics. (C) 2015 Elsevier Ltd. All rights reserved.