Thermal conductivity is a familiar property of materials: silver conducts heat well, and plastic does not. In recent years, an interdisciplinary group of materials scientists, engineers, physicists, and chemists have succeeded in pushing back long-established limits in the thermal conductivity of materials. Carbon nanotubes and graphene are at the high end of the thermal conductivity spectrum due to their high sound velocities and relative lack of processes that scatter phonons. Unfortunately, the superlative thermal properties of carbon nanotubes have not found immediate application in composites or interface materials because of difficulties in making good thermal contact with the nanotubes. At the low end of the thermal conductivity spectrum, solids that combine order and disorder in the random stacking of two-dimensional crystalline sheets, so-called “disordered layered crystals,” show a thermal conductivity that is only a factor of 2 larger than air. The cause of this low thermal conductivity may be explained by the large anisotropy in elastic constants that suppresses the density of phonon modes that propagate along the soft direction. Low-dimensional quantum magnets demonstrate that electrons and phonons are not the only significant carriers of heat. Near room temperature, the spin thermal conductivity of spin-ladders is comparable to the electronic thermal conductivities of metals. Our measurements of nanoscale thermal transport properties employ a variety of ultrafast optical pump-probe metrology tools that we have developed over the past several years. We are currently working to extend these techniques to high pressures (60 GPa), high magnetic fields (5 T), and high temperatures (1000 K).