We describe how hierarchical concepts can be represented in three types of layered neural networks. The aim is to support recognition of the concepts when partial information about the concepts is presented, and also when some of the neurons in the network might fail. Our failure model involves initial random failures. The three types of networks are: feed-forward networks with high connectivity, feed-forward networks with low connectivity, and layered networks with low connectivity and with both forward edges and "lateral" edges within layers. In order to achieve fault-tolerance, the representations all use multiple representative neurons for each concept. We show how recognition can work in all three of these settings, and quantify how the probability of correct recognition depends on several parameters, including the number of representatives and the neuron failure probability. We also discuss how these representations might be learned, in all three types of networks. For the feed-forward networks, the learning algorithms are similar to ones used in [4], whereas for networks with lateral edges, the algorithms are generally inspired by work on the assembly calculus [3, 6, 7].