Functional Dependencies (FDs) define attribute relationships based on syntactic equality, and, when usedin data cleaning, they erroneously label syntactically different but semantically equivalent values as errors. We explore dependency-based data cleaning with Ontology Functional Dependencies(OFDs), which express semantic attribute relationships such as synonyms and is-a hierarchies defined by an ontology. We study the theoretical foundations for OFDs, including sound and complete axioms and a linear-time inference procedure. We then propose an algorithm for discovering OFDs (exact ones and ones that hold with some exceptions) from data that uses the axioms to prune the search space. Towards enabling OFDs as data quality rules in practice, we study the problem of finding minimal repairs to a relation and ontology with respect to a set of OFDs. We demonstrate the effectiveness of our techniques on real datasets, and show that OFDs can significantly reduce the number of false positive errors in data cleaning techniques that rely on traditional FDs.