<p>The increasing energy and latency demands of artificial intelligence workloads in data-intensive applications are limited by the traditional von Neumann computer architecture, in which the memory and processing units are physically separated. Non-volatile memory technologies offer opportunities for analog in-memory computing that can go beyond these limits. These architectures consist of artificial synapses in dense crossbar arrays in which processing and memory can be combined. However, artificial synapses have various non-ideal effects, including nonlinearity, asymmetry, variability, limited endurance and retention loss which affect their capacity to be integrated into reliable accelerator platforms. In this Review, we consider the ideal synapse model and classify the effects that deviate from this ideality. In addition to discussing techniques to counter non-idealities that span material, device, circuit-level and algorithm-level design, we also consider cases that leverage deviations from the ideal towards specific applications. We make the case that the most effective synapse is one whose imperfections are well understood and can be harnessed in synergy with system-level goals.</p>