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This paper presents a framework for developing and executing parallel and distributed applications using the peer-to-peer computing model. The framework - called P2PComp - follows the main philosophy of the pure peer-to-peer model, since there is no hierarchy among the peers, all peers have the same functions and there is no central authority server responsible for the system organization. SPMD parallel applications can be implemented by extending the framework functionalities, which includes functions for starting and monitoring processes, searching resources and communicating by message passing. This paper presents a detailed description of the framework and examples of its utilization for building and executing parallel applications. The results obtained show that the framework can be effectively used for executing computational programs in a flexible peer-to-peer environment.