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Replicating human-like dexterity in robot hands represents one of the largest open problems in robotics. Reinforcement learning is a promising approach that has achieved impressive progress in the last few years; however, the class of problems it has typically addressed corresponds to a rather narrow definition of dexterity as compared to human capabilities. To address this gap, we investigate piano-playing, a skill that challenges even the human limits of dexterity, as a means to test high-dimensional control, and which requires high spatial and temporal precision, and complex finger coordination and planning. We introduce RoboPianist, a system that enables simulated anthropomorphic hands to learn an extensive repertoire of 150 piano pieces where traditional model-based optimization struggles. We additionally introduce an open-sourced environment, benchmark of tasks, interpretable evaluation metrics, and open challenges for future study. Our website featuring videos, code, and datasets is available at https://kzakka.com/robopianist/