A combinatorial framework for adversarial network coding is presented. Channels are described by specifying the possible actions that one or more (possibly coordinated) adversaries may take. Upper bounds on three notions of capacity (the one-shot capacity, the zero-error capacity, and the compound zero-error capacity) are obtained for point-to-point channels, and generalized to corresponding capacity regions appropriate for multi-source networks. A key result of this paper is a general method by which bounds on these capacities in point-to-point channels may be ported to networks. This technique is illustrated in detail for Hamming-type channels with multiple adversaries operating on specific coordinates, which correspond, in the context of networks, to multiple adversaries acting on specific network edges. Capacity-achieving coding schemes are described for some of the considered adversarial models.