Abstract: Defenses against adversarial examples, such as adversarial training, are typically tailored to a single perturbation type (e.g., small $\ell_\infty$-noise). For other perturbations, these defenses offer no guarantees and, at times, even increase the model's vulnerability. Our aim is to understand the reasons underlying this robustness trade-off, and to train models that are simultaneously robust to multiple perturbation types. We prove that a trade-off in robustness to different types of $\ell_p$-bounded and spatial perturbations must exist in a natural and simple statistical setting. We corroborate our formal analysis by demonstrating similar robustness trade-offs on MNIST and CIFAR10. We propose new multi-perturbation adversarial training schemes, as well as an efficient attack for the $\ell_1$-norm, and use these to show that models trained against multiple attacks fail to achieve robustness competitive with that of models trained on each attack individually. In particular, we find that adversarial training with first-order $\ell_\infty, \ell_1$ and $\ell_2$ attacks on MNIST achieves merely $50\%$ robust accuracy, partly because of gradient-masking. Finally, we propose affine attacks that linearly interpolate between perturbation types and further degrade the accuracy of adversarially trained models.