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Deep reinforcement learning (DRL) provides a model-agnostic approach to control complex dynamical systems, but has not been shown to scale to high-dimensional dexterous manipulation. Furthermore, deployment of DRL on physical systems remains challenging due to sample inefficiency. In this work, we show that model-free DRL with natural policy gradients can effectively scale up to complex manipulation tasks with a high-dimensional 24-DoF hand, and solve them from scratch in simulated experiments. We demonstrate successful policies for multiple complex tasks: object relocation, in-hand manipulation, tool use, and dooropening.