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An overview of the direct and ab initio simulation as well as experimental tools addressing heat conduction in dielectric nanostructures and nanostructured materials is proposed. Recent advancements based on density functional theory to compute phonon properties and thermal conductivity are exposed and the state of the art in metrology is detailed. Finally, new progresses in the experimental demonstration of coherence effects in heat conduction are illustrated, emphasizing the open questions in the still poorly understood field of phononic heat transport.