Understanding thermal transport processes can guide the rational design of devices and systems for thermal energy conversion and management. Despite the significant progress in thermal transport of inorganic crystals, thermal transport in complicated materials, such as polymers and hybrid materials, remains largely unexplored. This thesis first presents our discovery of the large thermal rectification effects in the novel tapered bottlebrush polymers using nonequilibrium molecular dynamic simulations. In sharp contrast to all other reported asymmetric nanostructures, we observed that the heat current from the wide end to the narrow end in tapered bottlebrush polymers is smaller than that in the opposite direction. It was demonstrated that a more disordered to less disordered structural transition within tapered bottlebrush polymers is essential for generating non-linearity in heat conduction for thermal rectification. Moreover, the thermal rectification factor increases with device length, reaching as high as ~70% with a device length of 28.5nm. This large thermal rectification with strong length dependence uncovers an unprecedented phenomenon – diffusive thermal transport in the forward direction and ballistic thermal transport in the backward direction. This thesis then focuses on thermal transport properties in hybrid materials: graphene-C60 heterostructures and hybrid organic-inorganic (CH3NH3)3Bi2I9 crystals. Graphene-C60 heterostructures assembled by van der Waals interactions between graphene and C60 have shown exciting potential for multifunctional devices. Understanding thermal transport in graphene-C60 heterostructures is the key to guiding the design of vdW heterostructures with desired thermal transport properties. Our equilibrium molecular dynamics simulations found that the in-plane thermal conductivity of the graphene-C60 heterostructure is as high as about 234 W/(mK) at room temperature, exceeding those of most pure metals. On the other hand, vdW interactions enhance the interfacial thermal conductance between graphene and C60 by strengthening out-of-plane phonon couplings between graphene and C60 and increasing in-plane and out-of-plane phonon couplings of the graphene layer. Our study demonstrates that the interfacial thermal conductance of graphene-C60 heterostructure is comparable to that of graphene-hexagonal boron-nitride (hBN) heterostructure. Hybrid perovskite analogues, such as methylammonium bismuth iodide (CH3NH3)3Bi2I9, have emerged as candidate photovoltaic and thermoelectric materials due to their low toxicity and high stability. Thermal transport and phonon properties of (CH3NH3)3Bi2I9 were studied neither experimentally nor theoretically, which hinders the optimal selection and design of stable, non-toxic hybrid perovskite material for photovoltaic and thermoelectric applications. We mapped out the phonon dispersion of (CH3NH3)3Bi2I9 single crystals at 300 K using inelastic x-ray scattering. The frequencies of acoustic phonons are among the lowest of crystals. Nanoindentation measurements verified that these crystals are very compliant and considerably soft. The frequency overlap between acoustic and optical phonons results in strong acoustic-optical scattering. All these features lead to an ultralow thermal conductivity.