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We are captivated by the promise of autonomous systems in our everyday life. However, ensuring that these systems act safely is an immense challenge: introducing complex systems into real-world uncertain environments while guaranteeing safety at all times is impossible in applications like self-driving vehicles, collaborative factory robots, and assistive robots. These systems will inevitably need to make real-time decisions with limited computational resources, and incomplete knowledge of the environment and other agents. This dissertation is an effort towards achieving trustworthy real-world autonomy by enabling autonomous systems to: (1) make theoretical safety guarantees efficiently based on known information, and(2) bridge the safety gap between this theory and the real world by reasoning about uncertainty in the environment and other agents.Towards this goal this dissertation covers various methods for scalable safety and real-time decision-making that draw from control theory, cognitive science, and learning, and are backed by both rigorous theory and physical testing on robotic platforms. We begin with an overview of reachability analysis and its applications for optimal control with safety guarantees. We then tackle the curse of dimensionality associated with reachability computation by decomposing systems or updating previous solutions in a warm-starting fashion. Next we explore planning in a simplified, low-dimensional space online with precomputed safety guarantees and tracking controls offline. This is extended to meta-planning, in which the online algorithm switches between faster/conservative modes and slower/accurate modes. Then we apply all of these tools towards navigating in uncertain environments among hard-to-predict agents such as human pedestrians. We use Bayesian machine learning methods to reason about human intention and navigate in a probabilistically safe manner. Finally, the dissertation ends with information about code bases and reachability tutorial examples.