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Wireless Mesh Networks (WMNs) have emerged as an important technology in building next generation fixed wireless broadband networks that provide low cost Internet access for fixed and mobile users. An orthogonal evolution in computer networking has been the rise of Peer-to-Peer (P2P) applications such as P2P file sharing. It is of interest to enable effective P2P file sharing in this type of networks. Our main contribution in this paper is innovative schemes for content caching and replication at mesh routers that enhance the performance of P2P file sharing in WMNs. We first motivate our proposed schemes by showing the impact of caching P2P content at mesh routers on the performance of P2P file sharing in WMNs. We then describe the design and operation of our content caching and replication schemes. Finally, we compare the performance of our proposed schemes against other existing schemes using simulations. We focus on P2P file sharing but other applications can also be supported by the proposed schemes.