Polymers are widely used in industry and in our daily life because of their diverse functionality, light weight, low cost and excellent chemical stability. However, on some applications such as heat exchangers and electronic packaging, the low thermal conductivity of polymers is one of the major technological barriers. Enhancing the thermal conductivity of polymers is important for these applications and has become a very active research topic over the past two decades. In this review article, we aim to: 1). systematically summarize the molecular level understanding on the thermal transport mechanisms in polymers in terms of polymer morphology, chain structure and inter-chain coupling; 2). highlight the rationales in the recent efforts in enhancing the thermal conductivity of nanostructured polymers and polymer nanocomposites. Finally, we outline the main advances, challenges and outlooks for highly thermal-conductive polymer and polymer nanocomposites. the inter-chain coupling. Fig. 1 Schematic diagrams of a polymer: (a) the morphology of a polymer consisting of crystalline and amorphous domains; (b) structure of a polymer chain.In addition to engineering the morphology of polymer chains, another common method to enhance the thermal conductivity of polymers is to blend polymers with highly thermal conductive fillers. The progress of nanotechnology over the last two decades not only provides more diverse high thermal conductivity fillers of different material types and topological shapes but also advances the understanding at the nanoscale. Fig. 2 shows a sketch of a polymer nanocomposite to illustrate the thermal transport mechanisms. In general, there are two types of polymer nanocomposites depending on whether nano-fillers form a network or not. When the filler concentration is low, no inter-filler networks could be formed, as shown in Fig. 2(a). The thermal conductivity is essentially determined by the filler-matrix coupling, i.e, interfacial thermal resistance, and the concentration and the geometric shapes of fillers. When the filler concentration is large enough, high conductivity fillers might form thermally conductive networks, as shown in Fig. 2(b). Although nanocomposites with filler network could possess a higher thermal conductivity than that without a network, their thermal conductivity could still be low due to the large inter-filler thermal contact resistance. Recently, three-dimensional fillers, such as carbon and graphene foams, have drawn a lot of attention. The fundamental thermal transport mechanisms and recent synthesis efforts in both types of nanocomposites are reviewed.This review article is organized as follows. In Section 2, we introduce the experimental progress on the enhancement of thermal conductivity by aligning polymer chains, and then review the methods to further tune the thermal conductivity by engineering chain structure and inter-chain coupling, as illustrated in Fig. 3(a). In Section 3, we discuss the thermal conductivity of polymer nanocomposites both with and without inter...