Autonomous vehicles operate with limited sensor horizons in unpredictable environments. To do so, they use a receding-horizon strategy to plan trajectories by executing a short plan while creating the next plan. Existing approaches to design trajectories for autonomous vehicles make a tradeoff between model complexity and planning speed, which can require sacrificing guarantees of safety and dynamic feasibility. This talk describes the Reachability-based Trajectory Design (RTD) method for trajectory planning. RTD begins with an offline Forward Reachable Set (FRS) computation of a robot's motion while it tracks parameterized trajectories; the FRS also provably bounds tracking error. At runtime, the FRS is used to map obstacles to the space of parameterized trajectories, which allows RTD to select a safe trajectory at every planning iteration. RTD prescribes a method of representing obstacles to ensure that these constraints can be created and evaluated in real time while maintaining provable safety. Persistent feasibility is achieved by prescribing a minimum duration of planned trajectories, and a minimum sensor horizon. The proposed method is shown as safe and persistently feasible across thousands of simulations and dozens of hardware demos.