Thermal properties play important roles in applications such as high efficiency thermoelectric materials and the thermal management of electronic devices. While electrons dominate thermal transport in metal, phonons (i.e., the quanta of lattice vibrations) are the primary energy carriers in crystalline insulators and semiconductors such as Silicon and Germanium. A fundamental understanding of the transport properties of multiple carriers including phonons and electrons can enable us to better design nanoscale materials. However, many of the current techniques to calculate the lattice thermal conductivity and interfacial thermal conductance involve major approximations.The first part of this work involves studying the role of scattering strength, such as anharmonicity (i.e., high order phonon-phonon scattering) and electron-phonon coupling, on thermal conductance at interfaces, based on Molecular Dynamics (MD) simulations. The second part of this work involves building accurate tools to predict the thermal conductivity of crystal materials. One approach is MD simulations. The other approach is lattice dynamics calculation from Boltzmann Transport Equations (BTE) combined with Fermi's golden rule, based on the force constants fitted from first-principles calculations. It is so far the most accurate approach and is essential for the prediction of thermal conductivity in crystal materials.With the completion of the tool for thermal conductivity prediction, several important semiconductor materials are studied using these first-principles methods. One is single layer molybdenum disulfide (MoS2), a two dimensional material with an intrinsic bandgap of 1.8 eV and a high electron mobility around 200 cm2V-1S-1, which can potentially enable applications in transistors, photovoltaics, valleytronics and thermoelectrics. Another material studied is Wurtzite Zinc-Oxide (w-ZnO), a wide bandgap semiconductor that holds promise in power electronics and transparent electronics applications. Finally, with the proven predictive power of our methods, we discover a strategy to tune thermal conductivity via bond engineering by modifying the bond saturation of materials. We find that penta-graphene and hydrogenated penta-graphene are promising candidates for future nanoelectronics applications due to their high lattice thermal conductivity. This helps providing a general guideline for the design of high thermal conductivity materials.All these studies provide us a better understanding of thermal transport in solid state crystal materials and can potentially enable us to design materials with desirable thermal transport properties.