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Deep neural networks (DNNs) are increasingly used in real-world applications (e.g. facial recognition). This has resulted in concerns about the fairness of decisions made by these models. Various notions and measures of fairness have been proposed to ensure that a decision-making system does not disproportionately harm (or benefit) particular subgroups of the population. In this paper, we argue that traditional notions of fairness that are only based on models' outputs are not sufficient when the model is vulnerable to adversarial attacks. We argue that in some cases, it may be easier for an attacker to target a particular subgroup, resulting in a form of \textit{robustness bias}. We show that measuring robustness bias is a challenging task for DNNs and propose two methods to measure this form of bias. We then conduct an empirical study on state-of-the-art neural networks on commonly used real-world datasets such as CIFAR-10, CIFAR-100, Adience, and UTKFace and show that in almost all cases there are subgroups (in some cases based on sensitive attributes like race, gender, etc) which are less robust and are thus at a disadvantage. We argue that this kind of bias arises due to both the data distribution and the highly complex nature of the learned decision boundary in the case of DNNs, thus making mitigation of such biases a non-trivial task. Our results show that robustness bias is an important criterion to consider while auditing real-world systems that rely on DNNs for decision making. Code to reproduce all our results can be found here: \url{https://github.com/nvedant07/Fairness-Through-Robustness}