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Safety is an essential component for deploying reinforcement learning (RL) algorithms in real-world scenarios, and is critical during the learning process itself. A natural first approach toward safe RL is to manually specify constraints on the policy's behavior. However, just as learning has enabled progress in large-scale development of AI systems, learning safety specifications may also be necessary to ensure safety in messy open-world environments where manual safety specifications cannot scale. Akin to how humans learn incrementally starting in child-safe environments, we propose to learn how to be safe in one set of tasks and environments, and then use that learned intuition to constrain future behaviors when learning new, modified tasks. We empirically study this form of safety-constrained transfer learning in three challenging domains: simulated navigation, quadruped locomotion, and dexterous in-hand manipulation. In comparison to standard deep RL techniques and prior approaches to safe RL, we find that our method enables the learning of new tasks and in new environments with both substantially fewer safety incidents, such as falling or dropping an object, and faster, more stable learning. This suggests a path forward not only for safer RL systems, but also for more effective RL systems.