Thermal conductivity is a basic and familiar property of materials that has played a pivotal role in condensed matter science for at least 150 years. In this talk I will emphasize recent examples of extreme behavior--and behavior under extreme conditions--where the physics is not well understood: we do not yet know the lower limit to the thermal conductivity of a fully dense material or the upper limit to the conductivity of a polymer or the mechanisms that limit thermal transport by spin excitations. Our measurements of heat conduction in novel materials are enabled by variety of ultrafast optical pump-probe metrology tools developed over the past decade. At the low end of the spectrum, solids that combine order and disorder in the random stacking of two-dimensional crystalline sheets show a thermal conductivity that is only a factor of 2 larger than air; the cause of this extreme behavior may be explained by the large anisotropy in elastic constants that suppresses the density of phonon modes that propagate along the soft direction. Extremes of high pressures (up to 60 GPa) allow us to continuous change phonon densities of states and phonon lifetimes to test theoretical predictions. The thermal conductivity of aligned, crystalline and liquid crystalline polymer fibers can be surprisingly high but not as high as needed for applications in thermal management. 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 pure metals.