The pace of improvement in the performance of conventional computer hardware has slowed significantly during the past decade, largely as a consequence of reaching the physical limits of manufacturing processes. To offset this slowdown, new approaches to HPC are now undergoing rapid development. This chapter describes current work on the development of cutting-edge exascale computing systems that are intended to be in place in 2021 and then turns to address several other important developments in HPC, some of which are only in the early stage of development. Domain-specific heterogeneous processing approaches use hardware that is tailored to specific problem types. Neuromorphic systems are designed to mimic brain function and are well suited to machine learning. And then there is quantum computing, which is the subject of some controversy despite the enormous funding initiatives that are in place to ensure that systems continue to scale-up from current small demonstration systems.