The thermal properties of carbon nanotubes are strongly dependent on their unique structure and size, and show promise as an ideal material for thermal management on the micro- and macro-scale. The specific heat of nanotubes is similar to that of two-dimensional graphene at high temperatures, but is sensitive to the effects of rolling the the graphene sheet into a small cylinder at low temperatures. Specifically, the acoustic phonon modes are stiffened due to the cylindrical geometry, and the phonon spectrum is quantized due to the small diameter of the tube. In bundles of single-walled nanotubes, the specific heat is a sensitive probe of inter-tube mechanical coupling. Measurements of the specific heat show that inter-tube coupling is relatively weak, and show direct evidence for quantum effects. The thermal conductivity of nanotubes should reflect the on-tube phonon structure. Aligned bundles of SWNTs show a high thermal conductivity (>200 W/m-K at room temperature), and possible quantization effects at low temperature.