This dissertation is focused on thermal transport at nanometer scale point and line constrictions and in nanowires with sawtooth surface roughness. To better understand thermal transport at a point contact such as that at the tip-sample junction of a scanning probe microscope, a Non Equilibrium Molecular Dynamics (NEMD) method is employed to calculate the temperature distribution and thermal resistance of a nanoscale point constriction formed between two silicon substrates. The simulation reveals surface reconstruction at the two free silicon surfaces and at the constriction. The radius of the heated zone in the cold substrate approaches a limit of about 20 times the average nearest-neighbor distance of boron doping atoms when the constriction radius (a) is reduced below the inter-dopant distance. The phonon mean free path at the constriction is suppressed by diffuse phonon-surface scattering and phonon-impurity scattering. The MD thermal resistance is close to the ballistic resistance when a is larger than 1 nm, suggesting that surface reconstruction does not reduce the phonon transmission coefficient significantly. When a is 0.5 nm and comparable to the dominant phonon wavelength, however, the NEMD result is considerably lower than the calculated ballistic resistance because bulk phonon dispersion and bulk potential are not longer accurate. The MD thermal resistance of the constriction increases slightly with increasing doping concentration due to the increase in the diffusive resistance. The NEMD method is further employed to calculate the temperature distribution and thermal resistance at nanoscale line constrictions formed between two silicone substrates. Similar to the nano point constriction, the thermal resistance at the nano line constriction is dominated by the ballistic resistance for constriction width in the range of 1 nm to 12 nm. An additional question that this dissertation seeks to answer is whether one can engineer the surface roughness on a nanowire to facilitate phonon backscattering so as to reduce the thermal conductivity below the diffuse surface limit. Monte Carlo simulation is used to show that phonon backscattering can occur at sawtooth surfaces of a silicon nanowire, suppressing the thermal conductivity below the diffuse surface limit. Asymmetric sawtooth nanowire surfaces can further cause phonon rectification, making the axial thermal conductance of the nanowire direction dependent. The phonon backscattering and rectification effects can be employed to enhance the thermoelectric figure of merit of nanowires. / text