In the conventional von Neumann (VN) architecture, data—both operands and operations to be performed on those operands—makes its way from memory to a dedicated central processor. With the end of Dennard scaling and the resulting slowdown in Moore’s law, the IT industry is turning its attention to non-Von Neumann (non-VN) architectures, and in particular, to computing architectures motivated by the human brain. One family of such non-VN computing architectures is artificial neural networks (ANNs). To be competitive with conventional architectures, such ANNs will need to be massively parallel, with many neurons interconnected using a vast number of synapses, working together efficiently to compute problems of significant interest. Emerging nonvolatile memories, such as phase-change memory (PCM) or resistive memory (RRAM), could prove very helpful for this, by providing inherently analog synaptic behavior in densely packed crossbar arrays suitable for on-chip learning. We discuss our recent research investigating the characteristics needed from such nonvolatile memory elements for implementation of high-performance ANNs. We describe experiments on a 3-layer perceptron network with 164,885 synapses, each implemented using 2 NVM devices. A variant of the backpropagation weight update rule suitable for NVM+selector crossbar arrays is shown and implemented in a mixed hardware–software experiment using an available, non-crossbar PCM array. Extensive tolerancing results are enabled by precise matching of our NN simulator to the conditions of the hardware experiment. This tolerancing shows clearly that NVM-based neural networks are highly resilient to random effects (NVM variability, yield, and stochasticity), but highly sensitive to gradient effects that act to steer all synaptic weights. Simulations of ANNs with both PCM and non-filamentary bipolar RRAM based on Pr1−xCaxMnO3 (PCMO) are also discussed. PCM exhibits smooth, slightly nonlinear partial-SET (conductance increase) behavior, but the asymmetry of its abrupt RESET introduces difficulties; in contrast, PCMO offers continuous conductance change in both directions, but exhibits significant nonlinearities (degree of conductance change depends strongly on absolute conductance). The quantitative impacts of these issues on ANN performance (classification accuracy) are discussed.