Title: Adversarially Robust Models may not Transfer Better: Sufficient Conditions for Domain Transferability from the View of Regularization. Abstract: Machine learning (ML) robustness and generalization are fundamentally correlated: they essentially concern about data distribution shift under adversarial and natural settings, respectively. Thus, it is critical to uncover their underlying connections to tackle one based on the other. On the one hand, recent studies show that more robust (adversarially trained) models are more generalizable to other domains. On the other hand, there lacks of theoretical understanding of such phenomenon and it is not clear whether there are counterexamples. In this paper, we aim to provide sufficient conditions for this phenomenon considering different factors that could affect both, such as the norm of last layer norm, Jacobian norm, and data augmentations (DA). In particular, we propose a general theoretical framework indicating factors that can be reformed as a function class regularization process, which could lead to the improvement of domain generalization. Our analysis, for the first time, shows that ``robustness" is actually not the causation for domain generalization; rather, robustness induced by adversarial training is a by-product of such function class regularization. We then discuss in details about different properties of DA and we prove that under certain conditions, DA can be viewed as regularization and therefore improve generalization. We conduct extensive experiments to verify our theoretical findings, and show several counterexamples where robustness and generalization are negatively correlated when the sufficient conditions are not satisfied.