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This article provides necessary and sufficient conditions for deadlock-free unicast and multicast routing with the path-based routing model in interconnection networks that use the wormhole switching technique. The theory is developed around three central concepts: channel waiting, False Resource Cycles, and valid destination sets. The first two concepts are suitable extensions to those developed for unicast routing by two authors of this article; the third concept has been developed by Lin and Ni. The necessary and sufficient conditions relax the requirements for deadlock-free routing, compared to techniques that provide only a sufficient condition. These necessary and sufficient conditions can be applied in a straightforward manner to prove deadlock freedom of newly developed adaptive routing algorithms for collective communication, which in turn will help in developing efficient and correct routing algorithms. The latter point is illustrated by developing two routing algorithms for multicast communication on 2D mesh architectures. The first algorithm uses fewer resources (channels) than an algorithm proposed in the literature but achieves the same adaptiveness. The second algorithm provides fully adaptive routing for both unicast and multicast messages.