Title: Rademacher Complexity Over $\mathcal{H} \Delta \mathcal{H}$ Class for Adversarially Robust Domain Adaptation. Abstract: In domain adaptation, a model is trained on a dataset generated from a source domain and its generalization is evaluated on a possibly different target domain. Understanding the generalization capability of the learned model is a longstanding question. Recent studies demonstrated that the adversarial robust learning under $\ell_\infty$ attack is even harder to generalize to different domains. To thoroughly study the fundamental difficulty behind adversarially robust domain adaptation, we propose to analyze a key complexity measure that controls the cross-domain generalization: the adversarial Rademacher complexity over $\mathcal{H} \Delta \mathcal{H}$ class. For linear models, we show that adversarial Rademacher complexity over $\mathcal{H} \Delta \mathcal{H}$ class is always greater than the non-adversarial one, which reveals the intrinsic hardness of adversarially robust domain adaptation. We also establish upper bounds on this complexity measure, and extend them to the ReLU neural network class as well. Finally, by properly extending our generalization bound for adversarially robust domain adaptation, we explain \emph{why adversarial training can help transferring the model performance to different domains}. We believe our results initiate the study of the generalization theory of adversarially robust domain adaptation, and could shed lights on distributed adversarially robust learning from heterogeneous sources -- a scenario typically encountered in federated learning applications.