In this paper we describe query processing strategies we have developed and implemented in the distributed version of the ORION object-oriented database system. The ORION query model is based on the ORION object-oriented data model. Further, we have adopted the response time as the primary objective function for query optimization. The query-processing strategies we have developed reflect our solutions to these requirements. In particular, our strategies are based on a dataflow execution model which represents a plan for executing a query concurrently at multiple sites. One important observation we bring out in our description of the ORION query-processing strategies is that most of the important techniques developed for optimizing and processing a relational query apply directly to an object-oriented query, despite the differences in the underlying data models.