In this article, we review thermal transport in polymers with different morphologies from aligned fibers to bulk amorphous states. We survey early and recent efforts in engineering polymers with high thermal conductivity by fabricating polymers with large-scale molecular alignments. The experimentally realized extremely high thermal conductivity of polymer nanofibers are highlighted, and understanding of thermal transport physics from molecular simulations are discussed. We then transition to the discussion of bulk amorphous polymers with an emphasize on the physics of thermal transport and its relation with the conformation of molecular chains in polymers. We also discuss the current understanding of how the chemistry of polymers would influence thermal transport in amorphous polymers and some limited, but important chemistry-structural-property relationships. Lastly, challenges, perspectives and outlook of this field are presented. We hope this review will inspire more fundamental and applied research in the polymer thermal transport field to advance scientific understanding and engineering applications.