A key challenge to the widespread deployment of robotic manipulators is the need to ensure safety in arbitrary environments while generating new motion plans in real-time. In particular, one must ensure that a manipulator does not collide with obstacles, collide with itself, or exceed its joint torque limits. This challenge is compounded by the need to account for uncertainty in the mass and inertia of manipulated objects, and potentially the robot itself. The present work addresses this challenge by proposing Autonomous Robust Manipulation via Optimization with Uncertainty-aware Reachability (ARMOUR), a provably-safe, receding-horizon trajectory planner and tracking controller framework for serial link manipulators. ARMOUR works by first constructing a robust, passivity-based controller that is proven to enable a manipulator to track desired trajectories with bounded error despite uncertain dynamics. Next, ARMOUR uses a novel variation on the Recursive Newton-Euler Algorithm (RNEA) to compute the set of all possible inputs required to track any trajectory within a continuum of desired trajectories. Finally, the method computes an over-approximation to the swept volume of the manipulator; this enables one to formulate an optimization problem, which can be solved in realtime, to synthesize provably-safe motion. The proposed method is compared to state of the art methods and demonstrated on a variety of challenging manipulation examples in simulation and on real hardware, such as maneuvering a dumbbell with uncertain mass around obstacles.