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The goal of the work described in this paper is to design and build a scalable infrastructure for executing grid applications on a widely distributed set of resources. Such grid infrastructure must be decentralized, robust, highly available, and scalable, while efficiently mapping application instances to available resources in the system. However, current desktop grid computing platforms are typically based on a client-server architecture, which has inherent shortcomings with respect to robustness, reliability and scalability. Fortunately, these problems can be addressed through the capabilities promised by new techniques and approaches in peer-to-peer (P2P) systems. By employing P2P services, our system allows users to submit jobs to be run in the system and to run jobs submitted by other users on any resources available in the system, essentially allowing a group of users to form an ad-hoc set of shared resources. The initial target application areas for the desktop grid system are in astronomy and space science simulation and data analysis.