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Robot motion planning is one of the central problems in robotics, and has received considerable amount of attention not only from roboticists but also from the control and artificial intelligence (AI) communities. Despite the different types of applications and physical properties of robotic systems, many high-level tasks of autonomous systems can be decomposed into subtasks which require point-to-point navigation while avoiding infeasible regions due to the obstacles in the workspace. This dissertation aims at developing a new class of sampling-based motion planning algorithms that are fast, efficient and asymptotically optimal by employing ideas from Machine Learning (ML) and Dynamic Programming (DP). First, we interpret the robot motion planning problem as a form of a machine learning problem since the underlying search space is not known a priori, and utilize random geometric graphs to compute consistent discretizations of the underlying continuous search space. Then, we integrate existing DP algorithms and ML algorithms to the framework of sampling-based algorithms for better exploitation and exploration, respectively. We introduce a novel sampling-based algorithm, called RRT#, that improves upon the well-known RRT* algorithm by leveraging value and policy iteration methods as new information is collected. The proposed algorithms yield provable guarantees on correctness, completeness and asymptotic optimality. We also develop an adaptive sampling strategy by considering exploration as a classification (or regression) problem, and use online machine learning algorithms to learn the relevant region of a query, i.e., the region that contains the optimal solution, without significant computational overhead. We then extend the application of sampling-based algorithms to a class of stochastic optimal control problems and problems with differential constraints. Specifically, we introduce the Path Integral - RRT algorithm, for solving optimal control of stochastic systems and the CL-RRT# algorithm that uses closed-loop prediction for trajectory generation for differential systems. One of the key benefits of CL-RRT# is that for many systems, given a low-level tracking controller, it is easier to handle differential constraints, so complex steering procedures are not needed, unlike most existing kinodynamic sampling-based algorithms. Implementation results of sampling-based planners for route planning of a full-scale autonomous helicopter under the Autonomous Aerial Cargo/Utility System Program (AACUS) program are provided.