Abstract: A century of experiment and theory have produced a thorough understanding of heat conduction by phonons in simple inorganic crystals.  By contrast, basic understanding of heat conduction by molecular vibrations in soft materials (amorphous and crystalline polymers, small molecule solids, and biological materials) is much less mature.  Complex, disordered structures spanning multiple length scales are difficult to characterize and model.  Low thermal conductivity, fiber morphologies, poor control of defects, and anisotropy created by molecular order create daunting challenges for experiment.   The lack of basic understanding of heat conduction in soft materials hinders applied research on the discovery of low, high, and switchable thermal conductivity materials for thermal insulation, thermal management, and heat exchangers.   I will discuss our work at U. Illinois on the thermal conductivity and elastic constants of a wide variety of polymeric materials in the form of thin films, fibers, and bulk materials, that span a factor of 300 in thermal conductivity, 0.06 to 20 W/m-K; and the high throughput experimental methods we are using to make our searches more efficient. Our most recent work emphasizes the changes in thermal conductivity that are produced by systematic variations in the molecular structure of epoxy thermosets and so-called “vitrimers”, polymer networks that incorporate dynamic covalent chemistry in the polymer network.  A liquid crystalline epoxy comprised of a twin-mesogen structure shows a dramatic odd-even effect where the isotropic thermal conductivity changes by a factor of 5, from 0.2 to 1.0 W/m-K, when the length of an alkane linker is changed from an even to an odd number of carbon atoms. Vitrimers constructed from precisely defined alkanes show a variation in thermal conductivity of a factor of 10, from 0.10 to 1.0 W/m-K as a function of crystallinity.