Across a wide range of domains, there is an urgent need for a well-founded approach to incorporating uncertain and incomplete knowledge into formal domain ontologies. Although this subject is receiving increasing attention from ontology researchers, there is as yet no broad consensus on the definition of a probabilistic ontology and on the most suitable approach to extending current ontology languages to support uncertainty. This paper presents two contributions to developing a coherent framework for probabilistic ontologies: (1) a formal definition of a probabilistic ontology, and (2) an extension of the OWL Web Ontology Language that is consistent with our formal definition. This extension, PR-OWL, is based on Multi-Entity Bayesian Networks (MEBN), a first-order Bayesian logic that unifies Bayesian probability with First-Order Logic. As such, PR-OWL combines the full representation power of OWL with the flexibility and inferential power of Bayesian logic.