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We consider opportunistic routing in wireless mesh networks. We exploit the inherent diversity of the broadcast nature of wireless by making use of multipath routing. We present a novel optimization framework for opportunistic routing based on network utility maximization (NUM) that enables us to derive optimal flow control, routing, scheduling, and rate adaptation schemes, where we use network coding to ease the routing problem. All previous work on NUM assumed unicast transmissions; however, the wireless medium is by its nature broadcast and a transmission will be received by multiple nodes. The structure of our design is fundamentally different; this is due to the fact that our link rate constraints are defined per broadcast region instead of links in isolation. We prove optimality and derive a primal-dual algorithm that lays the basis for a practical protocol. Optimal MAC scheduling is difficult to implement, and we use 802.11-like random scheduling rather than optimal in our comparisons. Under random scheduling, our protocol becomes fully decentralized (we assume ideal signaling). The use of network coding introduces additional constraints on scheduling, and we propose a novel scheme to avoid starvation. We simulate realistic topologies and show that we can achieve 20%-200% throughput improvement compared to single path routing, and several times compared to a recent related opportunistic protocol (MORE).