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We define a class of networks, called matroidal networks, which includes as special cases all scalar-linearly solvable networks, and in particular solvable multicast networks. We then present a method for constructing matroidal networks from known matroids. We specifically construct networks that play an important role in proving results in the literature, such as the insufficiency of linear network coding and the unachievability of network coding capacity. We also construct a new network, from the Vamos matroid, which we call the Vamos network, and use it to prove that Shannon-type information inequalities are in general not sufficient for computing network coding capacities. To accomplish this, we obtain a capacity upper bound for the Vamos network using a non-Shannon-type information inequality discovered in 1998 by Zhang and Yeung, and then show that it is smaller than any such bound derived from Shannon-type information inequalities. This is the first application of a non-Shannon-type inequality to network coding. We also compute the exact routing capacity and linear coding capacity of the Vamos network. Finally, using a variation of the Vamos network, we prove that Shannon-type information inequalities are insufficient even for computing network coding capacities of multiple-unicast networks.