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In this paper, we present an algorithm to build such controllers for physically simulated characters having many degrees of freedom. Our physics-based controllers are learned by using conditional VAEs, which can perform a variety of behaviors that are similar to motions in the training dataset. The controllers are robust enough to generate more than a few minutes of motion without conditioning on specific goals and to allow many complex downstream tasks to be solved efficiently. To show the effectiveness of our method, we demonstrate controllers learned from several different motion capture databases and use them to solve a number of downstream tasks that are challenging to learn controllers that generate natural-looking motions from scratch. We also perform ablation studies to demonstrate the importance of the elements of the algorithm. 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