In many real-world applications, test data may commonly exhibit categorical shifts, characterized by the emergence of novel classes, as well as distribution shifts, arising from feature distributions different from the ones the model was trained on. However, existing methods either discover novel classes in the open-world setting, or assume domain shift without an ability to discover novel classes. Here, we introduce cross-domain open-world discovery setting where the goal is to assign samples to seen classes and discover unseen classes under a domain shift. To address this challenging problem, we present CROW, a prototype based approach that introduces cluster-then-match strategy enabled by a well-structured representation space of foundation models. In this way, CROW discovers novel classes by robustly matching clusters with previously seen classes, followed by fine-tuning representation space with an objective designed for cross-domain open-world discovery. Extensive experimental results on image classification benchmark datasets demonstrate that CROW outperforms alternative baselines, achieving 8% performance improvement across 75 experimental settings.