Data matching - which decides whether two data elements (e.g., string, tuple, column, or knowledge graph entity) are the "same" (a.k.a. a match) - is a key concept in data integration, such as entity matching and schema matching. The widely used practice is to build task-specific or even dataset-specific solutions, which are hard to generalize and disable the opportunities of knowledge sharing that can be learned from different datasets and multiple tasks. In this paper, we propose Unicorn, a unified model for generally supporting common data matching tasks. Unicorn can enable knowledge sharing by learning from multiple tasks and multiple datasets, and can also support zero-shot prediction for new tasks with zero labeled matching/non-matching pairs. However, building such a unified model is challenging due to heterogeneous formats of input data elements and various matching semantics of multiple tasks. To address the challenges, Unicorn employs one generic Encoder that converts any pair of data elements (a, b) into a learned representation, and uses a Matcher, which is a binary classifier, to decide whether a matches b. To align matching semantics of multiple tasks, Unicorn adopts a mixture-of-experts model that enhances the learned representation into a better representation. We conduct extensive experiments using 20 datasets on seven well-studied data matching tasks, and find that our unified model can achieve better performance on most tasks and on average, compared with the state-of-the-art specific models trained for ad-hoc tasks and datasets separately. Moreover, Unicorn can also well serve new matching tasks with zero-shot learning.