Abstract: The conduction of heat in low-dimensional systems plays an important role in the thermal management of nanoscale devices, and is also a fundamental physical phenomenon in nature. In this work we address the fundamental physical aspects of thermal transport in various nanoscale materials, including nanotubes, nanowires and two-dimensional (2D) materials. We demonstrate that phonon transport in one-dimensional nanomaterials is superdiffusive, which leads to violation of Fourier's law. Moreover, using a core-shell nanowire as an example, the phonon coherent mechanism to reduce thermal conductivity is presented. Subsequently, we introduce recent theoretical and experimental studies of heat conduction in 2D materials.