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Data shift robustness has been primarily investigated from a fully supervised perspective, and robustness of zero-shot learning (ZSL) models have been largely neglected. In this paper, we present novel analyses on the robustness of discriminative ZSL to image corruptions. We subject several ZSL models to a large set of common corruptions and defenses. In order to realize the corruption analysis, we curate and release the first ZSL corruption robustness datasets SUN-C, CUB-C and AWA2-C. We analyse our results by taking into account the dataset characteristics, class imbalance, class transitions between seen and unseen classes and the discrepancies between ZSL and GZSL performances. Our results show that discriminative ZSL suffers from corruptions and this trend is further exacerbated by the severe class imbalance and model weakness inherent in ZSL methods. We then combine our findings with those based on adversarial attacks in ZSL, and highlight the different effects of corruptions and adversarial examples, such as the pseudo-robustness effect present under adversarial attacks. We also obtain new strong baselines for both models with the defense methods. Finally, our experiments show that although existing methods to improve robustness somewhat work for ZSL models, they do not produce a tangible effect.