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The ability to autonomously explore and navigate a physical space is a fundamental requirement for virtually any mobile autonomous agent, from household robotic vacuums to autonomous vehicles. Traditional SLAM-based approaches for exploration and navigation largely focus on leveraging scene geometry, but fail to model dynamic objects (such as other agents) or semantic constraints (such as wet floors or doorways). Learning-based RL agents are an attractive alternative because they can incorporate both semantic and geometric information, but are notoriously sample inefficient, difficult to generalize to novel settings, and are difficult to interpret. In this paper, we combine the best of both worlds with a modular approach that learns a spatial representation of a scene that is trained to be effective when coupled with traditional geometric planners. Specifically, we design an agent that learns to predict a spatial affordance map that elucidates what parts of a scene are navigable through active self-supervised experience gathering. In contrast to most simulation environments that assume a static world, we evaluate our approach in the VizDoom simulator, using large-scale randomly-generated maps containing a variety of dynamic actors and hazards. We show that learned affordance maps can be used to augment traditional approaches for both exploration and navigation, providing significant improvements in performance.