The theory of phonon transport in condensed matter is over 50 years old. However, the existence of classical size effects experimentally has been well established only in the last 10 years. With an outburst in the use of nanostructures for microelectronics and many energy related applications it has become necessary to dig deeper into fundamental aspects of phonon transport in nanomaterials in order to control their performance and engineer them for desired properties. In this thesis, the frequency and polarization dependent effects that are found to be particularly important to resolve for thermal transport in low dimensional and bulk nanostructures are addressed. Specifically two model nanostructures namely graphene and Si/Ge heterostructures are chosen due to their technological relevance. Phonon transport in both materials is simulated under the framework of the Boltzmann transport equation with an emphasis to resolve mode dependent effects during phonon-phonon scattering and phonon-boundary/interface scattering. It has been found that flexural phonons are the dominant groups that carry heat in single- and few-layer graphene due to their reduced anharmonic scattering. The inadequacy of the single mode relaxation time approximation to describe thermal transport and the peculiar frequency content that arises as a result of the strong coupling to non-equilibrium population is described in detail. Important decay mechanisms in single-layer graphene that are important for heat dissipation and optical absroption are presented. A quantitative understanding of decrease in thermal conductivity, extra scattering mechanisms and the transition to graphite in few-layer graphene is discussed. Through detailed multi-band frequency dependent relaxation time BTE simulations, we highlight the near interface phonon non-equilibrium that arises during phonon transport across Si/Ge interfaces and its effects on interfacial thermal resistance and thermal conductivity of superlattices/ nanocomposites. It is seen that interfaces and boundaries efficiently decrease the contribution of low frequency acoustic phonons but are less effective in decreasing those from mid-large frequencies. The simulations and mechanisms outlined in this thesis are expected to find widespread use for engineering high performance thermal interfaces, electronics and thermo-electrics.