Since early days of their discovery, it has been realized that Carbon Nanotubes (CNTs) have an unusually high thermal conductivity. Unfortunately, the amount of heat they can transfer from one medium to another can be limited by their thermal contact resistance, Rc, which in the worst case can result in thermally insulating bulk materials. Prior studies on individual nanotubes have reached various disparate conclusions, partly because many techniques employed for measuring such small samples rely on uncharacterized heat sources thus leaving fundamental uncertainties in the measurements. This has caused concerns that the true potential of these extraordinary thermal conductors will remain untapped. Relying on solid to liquid phase transition of sub-200nm Indium islands for thermometry, we report direct measurement of Rc by employing an independently characterized metallic heat source. Also we demonstrate that this contact resistance can be reduced by almost two orders of magnitude if a CNT is imbedded into metal contacts. Additionally in our preliminary data on a self-heated CNT we observe that the substrate gets hot while the CNT itself remains cold when electric current is passed through it. This observation cannot be explained by assuming joule heating to be the primary source of heat transfer. We can qualitatively explain these results by assuming that hot electrons flowing through the biased CNT can be scattered off the phonons of a dielectric substrate. Principles of the novel measurement technique, experimental results and simulations are presented in this dissertation.