We have used molecular dynamics to calculate the thermal conductivity of symmetric and asymmetric graphene nanoribbons (GNRs) of several nanometers in size (up to approximately 4 nm wide and approximately 10 nm long). For symmetric nanoribbons, the calculated thermal conductivity (e.g., approximately 2000 W/m-K at 400 K for a 1.5 nm x 5.7 nm zigzag GNR) is on the similar order of magnitude of the experimentally measured value for graphene. We have investigated the effects of edge chirality and found that nanoribbons with zigzag edges have appreciably larger thermal conductivity than nanoribbons with armchair edges. For asymmetric nanoribbons, we have found significant thermal rectification. Among various triangularly shaped GNRs we investigated, the GNR with armchair bottom edge and a vertex angle of 30 degrees gives the maximal thermal rectification. We also studied the effect of defects and found that vacancies and edge roughness in the nanoribbons can significantly decrease the thermal conductivity. However, substantial thermal rectification is observed even in the presence of edge roughness.