There is a significant need to build efficient non-von Neumann computing systems for highly data-centric artificial intelligence related applications. Brain-inspired computing is one such approach that shows significant promise. Memory is expected to play a key role in this form of computing and, in particular, phase-change memory (PCM), arguably the most advanced emerging non-volatile memory technology. Given a lack of comprehensive understanding of the working principles of the brain, brain-inspired computing is likely to be realized in multiple levels of inspiration. In the first level of inspiration, the idea would be to build computing units where memory and processing co-exist in some form. Computational memory is an example where the physical attributes and the state dynamics of memory devices are exploited to perform certain computational tasks in the memory itself with very high areal and energy efficiency. In a second level of brain-inspired computing using PCM devices, one could design a co-processor comprising multiple cross-bar arrays of PCM devices to accelerate the training of deep neural networks. PCM technology could also play a key role in the space of specialized computing substrates for spiking neural networks, and this can be viewed as the third level of brain-inspired computing using these devices.