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Peer-to-peer (P2P) networks and grids are distributed computing models that enable decentralized collaboration by integrating computers into networks in which each can consume and offer services. P2P is a class of self-organizing systems or applications that takes advantage of distributed resources storage, processing, information, and human presence available at the Internet's edges. A grid is a geographically distributed computation platform comprising a set of heterogeneous machines that users can access through a single interface. Both are hot research topics because they offer promising paradigms for developing efficient distributed systems and applications. Unlike the classic client-server model, in which roles are well separated, P2P and grid networks can assign each node a client or server role according to the operations they are to perform on the network - even if some nodes act more as server than as client in current implementations. In spite of current practices and thoughts, the grid and P2P models share several features and have more in common than we perhaps generally recognize. It is time to consider how to integrate these two models. A synergy between the two research communities, and the two computing models, could start with identifying the similarities and differences between them.