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Reinforcement learning (RL) has become a highly successful framework for learning in Markov decision processes (MDP). Due to the adoption of RL in realistic and complex environments, solution robustness becomes an increasingly important aspect of RL deployment. Nevertheless, current RL algorithms struggle with robustness to uncertainty, disturbances, or structural changes in the environment. We survey the literature on robust approaches to reinforcement learning and categorize these methods in four different ways: (i) Transition robust designs account for uncertainties in the system dynamics by manipulating the transition probabilities between states; (ii) Disturbance robust designs leverage external forces to model uncertainty in the system behavior; (iii) Action robust designs redirect transitions of the system by corrupting an agent’s output; (iv) Observation robust designs exploit or distort the perceived system state of the policy. Each of these robust designs alters a different aspect of the MDP. Additionally, we address the connection of robustness to the risk-based and entropy-regularized RL formulations. The resulting survey covers all fundamental concepts underlying the approaches to robust reinforcement learning and their recent advances.