This thesis explores the challenge of robustly handling dynamic obstacle uncertainty in autonomous driving systems. The path planning performance of Cornell's autonomous vehicle platform Skynet in the DARPA Urban Challenge (DUC) is analyzed and a new contingency planning formulation is presented that incorporates anticipated obstacle motions for improved collision avoidance capabilities. A discrete set of trajectory predictions is generated for each dynamic obstacle in the environment based on possible maneuvers the obstacle might make. A set of contingency paths is then optimized in real-time to accurately account for the mutually exclusive nature of these obstacle predictions. Computational scaling is addressed using a trajectory clustering algorithm that allows the contingency planner to plan a fixed number of paths regardless of the number of dynamic obstacles and possible obstacle goals in the environment. This contingency planning approach is evaluated using a series of human-inthe-loop experiments and simulations and is found to offer significant improvements in safety compared to the DUC planner and in performance compared to non-contingency planning approaches. A method for performing multi-step prediction over a two-stage Gaussian Process (GP) model is also presented. This prediction method is applied to a two-stage driver-vehicle obstacle model for the generation of high quality obstacle motion predictions using observed obstacle trajectories. An on-the-fly data selection technique is used to minimize computation when analytically evaluating higher order moments of the GP output. An adaptive Gaussian mixture model approach is also presented that allows this prediction technique to accurately predict the motion of highly nonlinear and multimodal systems.