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Heat conduction is an important energy transport process in nature. Phonon is the major energy carrier for heat in semiconductors and dielectric materials. In analogy to Ohm’s law of electrical conduction, Fourier’s law is the fundamental law of heat conduction in solids. Although Fourier’s law has received great success in describing macroscopic heat conduction in the past two hundred years, its validity in low dimensional systems is still an open question. Here we give a brief review of the recent developments in experimental, theoretical and numerical studies of heat conduction in low dimensional systems, including lattice models and low dimensional nanostructures such as nanowires, nanotubes and graphene. We will demonstrate that phonons transport in low dimensional systems superdiffusively, which leads to a size dependent thermal conductivity. In other words, Fourier’s law is not applicable in …