The SemanticWeb envisions a World Wide Web in which data is described with rich semantics and applications can pose complex queries. To this point, researchers have defined new languages for specifying meanings for concepts and developed techniques for reasoning about them, using RDF as the data model. To flourish, the Semantic Web needs to provide interoperability -- both between sites with different terminologies and with existing data and the applications operating on them. To achieve this, we are faced with two problems. First, most of the world's data is available not in RDF but in XML; XML and the applications consuming it rely not only on the domain structure of the data, but also on its document structure. Hence, to provide interoperability between such sources, we must map between both their domain structures and their document structures. Second, data management practitioners often prefer to exchange data through local point-to-point data translations, rather than mapping to common mediated schemas or ontologies. This paper describes the Piazza system, which addresses these challenges. Piazza offers a language for mediating between data sources on the SemanticWeb, and it maps both the domain structure and document structure. Piazza also enables interoperation of XML data with RDF data that is accompanied by rich OWL ontologies. Mappings in Piazza are provided at a local scale between small sets of nodes, and our query answering algorithm is able to chain sets mappings together to obtain relevant data from across the Piazza network. We also describe an implemented scenario in Piazza and the lessons we learned from it.