The human visual system is remarkably robust against a wide range of naturally occurring variations and corruptions like rain or snow. In contrast, the performance of modern image recognition models strongly degrades when evaluated on previously unseen corruptions. Here, we demonstrate that a simple but properly tuned training with additive Gaussian and Speckle noise generalizes surprisingly well to unseen corruptions, easily reaching the state of the art on the corruption benchmark ImageNet-C (with ResNet50) and on MNIST-C. We build on top of these strong baseline results and show that an adversarial training of the recognition model against locally correlated worst-case noise distributions leads to an additional increase in performance. This regularization can be combined with previously proposed defense methods for further improvement.