Abstract: Several existing works study either adversarial or natural distributional robustness of deep neural networks separately. In practice, however, models need to enjoy both types of robustness to ensure reliability. In this work, we bridge this gap and show that in fact, {\it explicit tradeoffs} exist between adversarial and natural distributional robustness. We first consider a simple linear regression setting on Gaussian data with disjoint sets of \emph{core} and \emph{spurious} features. In this setting, through theoretical and empirical analysis, we show that (i) adversarial training with $\ell_1$ and $\ell_2$ norms increases the model reliance on spurious features; (ii) For $\ell_\infty$ adversarial training, spurious reliance only occurs when the scale of the spurious features is larger than that of the core features; (iii) adversarial training can have {\it an unintended consequence} in reducing distributional robustness, specifically when spurious correlations are changed in the new test domain. Next, we present extensive empirical evidence, using a test suite of twenty adversarially trained models evaluated on five benchmark datasets (ObjectNet, RIVAL10, Salient ImageNet-1M, ImageNet-9, Waterbirds), that adversarially trained classifiers rely on backgrounds more than their standardly trained counterparts, validating our theoretical results. We also show that spurious correlations in training data (when preserved in the test domain) can {\it improve} adversarial robustness, revealing that previous claims that adversarial vulnerability is rooted in spurious correlations are incomplete.