A great deal of research in recent years has focused on robot learning.  In many applications, guarantees that specifications are satisfied throughout the learning process are paramount. For the safety specification, we present a controller synthesis technique based on the computation of reachable sets, using optimal control and game theory.  We present new methods for computing the reachable set, based on a functional approximation which has the potential to broadly alleviate its computational complexity.  In the second part of the talk, we will present a toolbox of methods combining reachability with data-driven techniques inspired by machine learning, to enable performance improvement while maintaining safety. We will illustrate these "safe learning" methods on robotic platforms at Berkeley, including demonstrations of motion planning around people, and navigating in a priori unknown environments.  

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