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Open-set recognition and adversarial defense study two key aspects of deep learning that are vital for real-world deployment. The objective of open-set recognition is to identify samples from open-set classes during testing, while adversarial defense aims to defend the network against images with imperceptible adversarial perturbations. In this paper, we show that open-set recognition systems are vulnerable to adversarial attacks. Furthermore, we show that adversarial defense mechanisms trained on known classes do not generalize well to open-set samples. Motivated by this observation, we emphasize the need of an Open-Set Adversarial Defense (OSAD) mechanism. This paper proposes an Open-Set Defense Network (OSDN) as a solution to the OSAD problem. The proposed network uses an encoder with feature-denoising layers coupled with a classifier to learn a noise-free latent feature representation. Two techniques are employed to obtain an informative latent feature space with the objective of improving open-set performance. First, a decoder is used to ensure that clean images can be reconstructed from the obtained latent features. Then, self-supervision is used to ensure that the latent features are informative enough to carry out an auxiliary task. We introduce a testing protocol to evaluate OSAD performance and show the effectiveness of the proposed method in multiple object classification datasets. The implementation code of the proposed method is available at: https://github.com/rshaojimmy/ECCV2020-OSAD.