Adversarial robustness research primarily focuses on L_p perturbations, and most defenses are developed with identical training-time and test-time adversaries. However, in real-world applications developers are unlikely to have access to the full range of attacks or corruptions their system will face. Furthermore, worst-case inputs are likely to be diverse and need not be constrained to the L_p ball. To narrow in on this discrepancy between research and reality we introduce ImageNet-UA, a framework for evaluating model robustness against a range of unforeseen adversaries, including eighteen new non-L_p attacks. To perform well on ImageNet-UA, defenses must overcome a generalization gap and be robust to a diverse attacks not encountered during training. In extensive experiments, we find that existing robustness measures do not capture unforeseen robustness, that standard robustness techniques are beat by alternative training strategies, and that novel methods can improve unforeseen robustness. We present ImageNet-UA as a useful tool for the community for improving the worst-case behavior of machine learning systems.