Polymers with high thermal conductivities are of great interest to both scientific research and industrial applications. In this dissertation, I use molecular dynamic (MD) simulations to study the fundamentals of thermal transport in bulk amorphous polymers, including 1) polymer blends, 2) block copolymers and 3) polyelectrolytes. In polymer blends, the thermal conductivity is strongly related to the chain conformation, especially the spatial extent of the polymer chains, which can be characterized by their radii of gyration (Rg). Increasing the inter-chain interactions in polymer blends will increase the thermal conductivity, which is due to the conformation change in the major component that eventually leads to an enhancement in thermal transport along the polymer chain backbone through the intra-chain bonding interactions. The polymer chain conformation and thermal conductivity relationship is also applicable to polyethylene-polypropylene (PE-PP) diblock copolymers. The thermal conductivity of the PE-PP diblock copolymer can be tuned continuously by the block ratio, which influences the Rg. However, in polyelectrolytes the mechanism of thermal conductivity enhancement with ionization ratio increasing is not due to the chain conformation change. In solid polyelectrolytes the chains are collapsed, and the thermal conductivity contribution from intra-molecular bonding interaction is constant. By adding metal cations, the Coulombic force attracts the atoms and increases the Lennard-Jones (LJ) repulsive force, and as a result the thermal conductivity contribution from LJ interaction increases. The fundamental structure- property relationship in soft materials (polymers) established in this research project can provide guidance for designing and synthesizing polymers with desirable thermal conductivity.The interfacial thermal conductance (ITC) between hard-soft interfaces is another critical factor that can affect the thermal conductivity of composite materials. I use 1) equilibrium molecular dynamics (EMD) simulations combined with the Green-Kubo (GK) formula and 2) phonon wave packet (WP) MD methods to study how different types of self-assembled monolayers (SAM) affect the thermal transport across the Au (gold)-SAM-organic liquid interfaces. In the EMD method, we focus on a practically synthesizable heterogeneous SAM structures with alternating short-long molecular chains, namely the "molecular fin" structure. The "molecular fin" structure is found to improve the thermal conductance by 46-68% compared to a homogeneous SAM. I find that the root reason of this enhancement in thermal conductance is due to the penetration of the liquid molecules into the spaces between the long SAM molecules, which increases the effective contact area. The WP method shows that the Au-COOH SAM-hexamine interface has the highest energy transmission coefficients of all phonons, and as a result its ITC is also the highest. The fundamental understanding of thermal transport across hard-soft interfaces can help design engineered surfaces with controlled thermal resistance.