Bulk polymers are generally regarded as thermal insulators, and typically have thermal conductivities on the order of 0.1 W m(-1) K(-1). However, recent work suggests that individual chains of polyethylene--the simplest and most widely used polymer--can have extremely high thermal conductivity. Practical applications of these polymers may also require that the individual chains form fibres or films. Here, we report the fabrication of high-quality ultra-drawn polyethylene nanofibres with diameters of 50-500 nm and lengths up to tens of millimetres. The thermal conductivity of the nanofibres was found to be as high as approximately 104 W m(-1) K(-1), which is larger than the conductivities of about half of the pure metals. The high thermal conductivity is attributed to the restructuring of the polymer chains by stretching, which improves the fibre quality toward an 'ideal' single crystalline fibre. Such thermally conductive polymers are potentially useful as heat spreaders and could supplement conventional metallic heat-transfer materials, which are used in applications such as solar hot-water collectors, heat exchangers and electronic packaging.