While a large amount of work has focused on designing adversarial attacks against image classifiers, only a few methods exist to attack semantic segmentation models. We show that attacking segmentation models presents task-specific challenges, for which we propose novel solutions. Our final evaluation protocol outperforms existing methods, and shows that those can overestimate the robustness of the models. Additionally, so far adversarial training, the most successful way for obtaining robust image classifiers, could not be successfully applied to semantic segmentation. We argue that this is because the task to be learned is more challenging, and requires significantly higher computational effort than for image classification. As a remedy, we show that by taking advantage of recent advances in robust ImageNet classifiers, one can train adversarially robust segmentation models at limited computational cost by fine-tuning robust backbones.