The performance of a reinforcement learning algorithm can vary drastically during learning because of exploration. Existing algorithms provide little information about the quality of their current policy before executing it, and thus have limited use in high-stakes applications, such as healthcare. We address this lack of accountability by proposing that algorithms output policy certificates. These certificates bound the sub-optimality and return of the policy in the next episode, allowing humans to intervene when the certified quality is not satisfactory. We further introduce two new algorithms with certificates and present a new framework for theoretical analysis that guarantees the quality of their policies and certificates. For tabular MDPs, we show that computing certificates can even improve the sample-efficiency of optimism-based exploration. As a result, one of our algorithms achieves regret and PAC bounds that are minimax up to lower-order terms.