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Fairness and robustness are two important notions of learning models. Fairness ensures that models do not disproportionately harm (or benefit) some groups over others, while robustness measures the models' resilience against small input perturbations. While equally important properties, this paper illustrates a dichotomy between fairness and robustness, and analyzes when striving for fairness decreases the model robustness to adversarial samples. The reported analysis sheds light on the factors causing such contrasting behavior, suggesting that distance to the decision boundary across groups as a key factor. Experiments on non-linear models and different architectures validate the theoretical findings. In addition to the theoretical analysis, the paper also proposes a simple, yet effective, solution to construct models achieving good tradeoffs between fairness and robustness.