Training agents to autonomously learn how to use anthropomorphic robotic hands has the potential to lead to systems capable of performing a multitude of complex manipulation tasks in unstructured and uncertain environments. In this work, we first introduce a suite of challenging simulated manipulation tasks that current reinforcement learning and trajectory optimisation techniques find difficult. These include environments where two simulated hands have to pass or throw objects between each other, as well as an environment where the agent must learn to spin a long pen between its fingers. We then introduce a simple trajectory optimisation that performs significantly better than existing methods on these environments. Finally, on the challenging PenSpin task we combine sub-optimal demonstrations generated through trajectory optimisation with off-policy reinforcement learning, obtaining performance that far exceeds either of these approaches individually, effectively solving the environment. Videos of all of our results are available at: https://dexterous-manipulation.github.io/