Neuromorphic – or brain-inspired – computing is a multi-disciplinary field of research aimed at extending our computational capabilities to tackle traditionally difficult problems, including perception, decision-making, prediction, and sensorimotor control. There is added urgency with the simultaneously decreasing benefits of CMOS scaling and increasing data processing demands. Along with new neuromorphic architectures and algorithms, an important area of research goes down to the device level to attempt to mimic neural functions. There are a number of emerging device technologies that may be attractive candidates for this functionality, including memristors. This tutorial will survey the device level concepts and properties of memristors and how they can be applied to building future brain-inspired computing systems. Topics covered include the conceptual requirements for mimicking the nervous system with some of the open questions. Chua’s local activity principle will be introduced, how it underpins the generation of spiking behavior in neurons, and some physical realizations. Various examples of artificial neural networks and their implementations with emerging devices will be surveyed, including recurrent and convolutional neural networks, perceptrons, Hopfield networks, and associative memories