Crystals and glasses exhibit fundamentally different heat conduction mechanisms: the periodicity of crystals allows for the excitation of propagating vibrational waves that carry heat, as first discussed by Peierls in 1929, while in glasses the lack of periodicity breaks Peierls' picture and heat is mainly carried by the coupling of vibrational modes, often described by a harmonic theory introduced by Allen and Feldman in 1989.

In this thesis we derive a unified microscopic equation that describes on an equal footing heat conduction in crystals, glasses, and anything in-between. In particular, such an equation predicts correctly and in agreement with experiments the thermal conductivity in crystals, glasses, and most importantly in the mixed regime of complex crystals with glasslike conductivity. This formulation is relevant for several technological applications, as it will potentially allow to predict and engineer the ultralow thermal conductivity of, for example, target materials for thermoelectric energy conversion and for thermal barrier coatings.

We also show how in the crystalline regime such a microscopic transport equation can be coarse grained into a set of "viscous heat equations", which generalize the macroscopic heat equation formulated by Fourier in 1822. These viscous heat equations account for both diffusion and heat hydrodynamics, and rationalize the recent discovery of heat transfer via temperature waves in graphitic devices.