Modern electronic devices are characterized by high-power and high-frequency with excessive heat accumulation. Thermal interface materials (TIMs) are of crucial importance for efficient heat dissipation to maintain proper functions and lifetime for these devices. The most promising TIMs are those polymer-based nanocomposites consisting of polymers and low-dimensional materials with high thermal conductivity (TC). This perspective summarizes the recent progress on the thermal transport properties of newly discovered one-dimensional (1D) nanomaterials and two-dimensional (2D) nanomaterials as well as three-dimensional (3D) nanostructures consisting of these 1D and 2D nanomaterials. Moreover, the applications of various nanomaterials in polymer nanocomposites for advanced TIMs are critically reviewed and the mechanism of TC enhancement is analysed. It is hoped that the present review could provide better understanding of the thermal transport properties of recently developed 2D nanomaterials and various 3D nanostructures as well as relevant polymer-based TIMs, shedding more light on the thermal management research.