Despite their impressive performance on image classification tasks, deep networks have a hard time generalizing to unforeseen corruptions of their data. To fix this vulnerability, prior works have built complex data augmentation strategies, combining multiple methods to enrich the training data. However, introducing intricate design choices or heuristics makes it hard to understand which elements of these methods are indeed crucial for improving robustness. In this work, we take a step back and follow a principled approach to achieve robustness to common corruptions. We propose PRIME, a general data augmentation scheme that relies on simple yet rich families of max-entropy image transformations. PRIME outperforms the prior art in terms of corruption robustness, while its simplicity and plug-and-play nature enable combination with other methods to further boost their robustness. We analyze PRIME to shed light on the importance of the mixing strategy on synthesizing corrupted images, and to reveal the robustness-accuracy trade-offs arising in the context of common corruptions. Finally, we show that the computational efficiency of our method allows it to be easily used in both on-line and off-line data augmentation schemes. Our code is available at https://github.com/amodas/PRIME-augmentations.