Semiconductor silicon (Si) is one of the best materials for microelectronic/ optoelectronic and photovoltaic device applications and has thus become the cornerstone of the semiconductor industry. Low-dimensional Si nanostructures are being increasingly adopted in various Si-based technologies, attracting great attention to their thermal transport properties. In Si-based materials, thermal energy is predominantly transported by phonons (quantized lattice vibrations). Although the reduction in thermal conductivity is a challenge with respect to the thermal management of nanoelectronic devices used in computer processors, it can be beneficial for thermoelectric (TE) energy conversion. A deeper understanding of thermal transport at the nanoscale is strategically important to achieve the optimal use of low-dimensional Si structures in emerging applications. In this work, we systematically investigate the phonon thermal transport in Si nanostructures, including zero-dimensional nanoclusters, one-dimensional nanowires, and two-dimensional nanosheets, through the use of the equilibrium molecular dynamics method. We first study the structural transition and phonon thermal conductivity of Si nanoclusters with different diameters. When the diameter of the cluster increases from 1.80 nm to 3.46 nm, the cluster structure changes from an amorphous state to a crystalline state, which is consistent with the reported experimental result. The calculated thermal conductivity of the Si nanoclusters is two to three orders of magnitude lower than that of bulk Si. In addition, size- and temperature-dependent effects on the thermal conductivity of the Si nanoclusters are also observed because of the remarkable phonon-boundary scattering and phonon-phonon scattering, respectively. For the one-dimensional nanostructured system, we focus on the effects of surface hydrogenation and nitrogenation on the phonon thermal conductivity of Si nanowires (SiNWs) at room temperature. We find that the phonon thermal conductivity of SiNWs is approximately two orders of magnitude lower than that of bulk Si, and it can be significantly affected by surface functionalization. Surface hydrogen passivation can saturate the dangling bonds and reduce the lattice mismatch between the inner and surface layers of SiNWs, thereby increasing the thermal conductivity to some degree compared with that of pure SiNWs without surface passivation. However, surface nitrogen passivation can significantly reduce the thermal conductivity. In particular, 50% surface nitrogenation on SiNWs can induce thermal conductivity attenuation by approximately 75% compared with that of fully hydrogenated SiNWs. This reduction in the thermal conductivity arises mainly from phonon scattering due to defects near the surface, as well as the suppression of some vibrational modes due to surface nitrogenation. Our simulations clearly demonstrate the importance of surface chemistry or functionalization in tuning the thermal conductivity, which has profound implications for TE applications of SiNWs. We also investigate the phonon thermal conductivity of two-dimensional, graphene-like silicene sheets at room temperature and find that the in-plane thermal conductivity of silicene sheets is approximately one order of magnitude lower than that of bulk Si. We further determine the effects of vacancy defects on the thermal conductivity and observe their significant diminution due to the effect of phonon-defect scattering; the underlying physical mechanism is explained from the phonon spectral analysis. Our results show that phonon transport in silicene sheets is strongly affected by vacancy concentration, vacancy size, and vacancy boundary shape. These findings can be used to guide the defect engineering of the thermal properties of low-dimensional Si materials. This thesis could be helpful in furthering the understanding of phonon thermal transport in low-dimensional Si nanostructures and may serve as a highly useful experimental guide in Si-based TE applications as well as in other thermal-related applications.