Polymers are large molecules composed of many repeating subunits. They widely exist as both natural biopolymers and synthetic plastics. The thermal transport properties of polymers are critical in many applications. For example, high thermal conductivity is desired for a wide variety of heat transfer applications, such as the thermal interface material and the heat spreader in microelectronic devices. Traditional polymers are found to have very low thermal conductivities, usually between 0.1 to 1 W/mK, because the defects such as polymer chain ends, entanglement, random orientation, voids, impurities and etc. act as stress concentration points and phonon scattering sites for heat transfer.Ultra drawing the polymers into nanofiber form with highly oriented chains can help eliminate these defects and improve the thermal conductivity. We use molecular dynamic simulations to study the structural origin of high thermal conductivity of these polymers in the nanofiber form, and unveil its relationship with thermal stability at the molecular level. Polymers with strong backbone and rigid backbone bonds, e.g. π-conjugated polymers, are found to have both high thermal conductivity and good thermal stability. On the other hand, the phase transition at ~400 K in polyethylene is found to cause signification thermal conductivity drop by as much as one order of magnitude. Although the phase transition in polyethylene questions its ability for high temperature applications, the phase transition and the high contrast thermal conductivity switch are found fully reversible. This makes it possible to use cheap and readily available polymers for thermal conductivity regulation in a wide range of applications such as thermal management of electronics, phononics, sensors, and energy storage and conversion.Another route to improve the thermal conductivity of polymer is through compositing, where the thermal transport across the interfaces between polymers and solid filler particles is important. Two factors are known to significantly impact the thermal transport efficiency across the interfaces: vibration coupling and interfacial adhesion strength. Our present results show that vibration coupling alone can improve the interfacial thermal conductance at gold-polyethylene interface by a factor of ~7. Our further investigation reports the ultra-efficient thermal transport across non-aqueous hydrogen bonds by molecular simulations of the thermal conductance across hard-soft material interfaces under systematically controlled interfacial bonding conditions. We found that these non-aqueous hydrogen bonds can enhance the thermal transport by as much as one order of magnitude.Besides fillers and interfaces, the intrinsic low thermal conductivity polymer matrix limits further improvement of thermally conductive composite materials. As polymer chains exhibit unique anisotropic molecular structure along covalently bonded backbone and across tunable inter-chains, detailed investigation of thermal transport in amorphous polymer by decomposing the effective thermal conductivity to different contributions from intra-chain and inter-chain interactions is provided. We find that the thermal transport in amorphous polymers with weak inter-chain interaction is dominated by along-chain heat conduction, and thermal conductivity is a strong function of the radius of gyration of the molecular chains. Polymers with stiffer molecular backbones have larger persistence length, which can lead to more extended chain morphology and thus higher thermal conductivity. In all, this dissertation sets understanding polymer materials with high thermal conductivity as the overarching goal. It provides comprehensive understanding of a variety of possible strategies to achieve this goal from orientated fibers via ultra drawing process, amorphous bulk with larger persistence length, to composite materials with enhanced thermal transport across hard/soft materials.