Great progress has been achieved in domain adaptation in decades. Existing works are always based on an ideal assumption that testing target domains are independent and identically distributed with training target domains. However, due to unpredictable corruptions (e.g., noise and blur) in real data, such as web images and real-world object detection, domain adaptation methods are increasingly required to be corruption robust on target domains. We investigate a new task, corruption-agnostic robust domain adaptation (CRDA), to be accurate on original data and robust against unavailable-for-training corruptions on target domains. This task is non-trivial due to the large domain discrepancy and unsupervised target domains. We observe that simple combinations of popular methods of domain adaptation and corruption robustness have suboptimal CRDA results. We propose a newapproach based on two technical insights into CRDA, as follows: (1) an easy-to-plug module called domain discrepancy generator (DDG) that generates samples that enlarge domain discrepancy to mimic unpredictable corruptions; (2) a simple but effective teacher-student scheme with contrastive loss to enhance the constraints on target domains. Experiments verify that DDG maintains or even improves its performance on original data and achieves better corruption robustness than baselines. Our code is available at: https://github.com/YifanXu74/CRDA.