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Patch-based adversarial attacks introduce a perceptible but localized change to the input that induces misclassification. While progress has been made in defending against imperceptible attacks, it remains unclear how patch-based attacks can be resisted. In this work, we study two different approaches for defending against black-box patch attacks. First, we show that adversarial training, which is successful against imperceptible attacks, has limited effectiveness against state-of-the-art location-optimized patch attacks. Second, we find that compositional deep networks, which have part-based representations that lead to innate robustness to natural occlusion, are robust to patch attacks on PASCAL3D+ and the German Traffic Sign Recognition Benchmark, without adversarial training. Moreover, the robustness of compositional models outperforms that of adversarially trained standard models by a large margin. However, on GTSRB, we observe that they have problems discriminating between similar traffic signs with fine-grained differences. We overcome this limitation by introducing part-based finetuning, which improves fine-grained recognition. By leveraging compositional representations, this is the first work that defends against black-box patch attacks without expensive adversarial training. This defense is more robust than adversarial training and more interpretable because it can locate and ignore adversarial patches.