This paper tackles the problem of computing views of OWL ontologies using a forgetting-based approach. In traditional relational databases, a view is a subset of a database, whereas in ontologies, a view is more than a subset; it contains not only axioms contained in the original ontology, but may also contain newly-derived axioms entailed by the original ontology (implicitly contained in the original ontology). Specifically, given an ontology O, the signature sig(O) of O is the set of all the names in O, and a view V of O is a new ontology obtained from O using only part of O's signature, namely the target signature, while preserving all logical entailments up to the target signature. Computing views of OWL ontologies is useful for Semantic Web applications such as ontology-based query answering, in a way that the view can be used as a substitute of the original ontology to answer queries formulated with the target signature, and information hiding, in the sense that it restricts users from viewing certain information of an ontology. Forgetting is a form of non-standard reasoning concerned with eliminating from an ontology a subset of its signature, namely the forgetting signature, in such a way that all logical entailments are preserved up to the target signature. Forgetting can thus be used as a means for computing views of OWL ontologies - the solution of forgetting a set F of names from an ontology O is the view V of O for the target signature sig(O)\F. In this paper, we present a forgetting-based method for computing views of OWL ontologies specified in the description logic ALCHOI, the basic ALC extended with role hierarchy, nominals and inverse roles. The method is terminating and sound. Despite the method not being complete, an evaluation with a prototype implementation of the method on a corpus of real-world ontologies has shown very good success rates. This is very useful from the perspective of the Semantic Web, as it provides knowledge engineers with a powerful tool for creating views of OWL ontologies.