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The advent of dedicated Deep Learning (DL) accelerators and neuromorphic processors has brought on new opportunities for applying both Deep and Spiking Neural Network (SNN) algorithms to healthcare and biomedical applications at the edge. This can facilitate the advancement of medical Internet of Things (IoT) systems and Point of Care (PoC) devices. In this paper, we provide a tutorial describing how various technologies including emerging memristive devices, Field Programmable Gate Arrays (FPGAs), and Complementary Metal Oxide Semiconductor (CMOS) can be used to develop efficient DL accelerators to solve a wide variety of diagnostic, pattern recognition, and signal processing problems in healthcare. Furthermore, we explore how spiking neuromorphic processors can complement their DL counterparts for processing biomedical signals. The tutorial is augmented with case studies of the vast literature on neural network and neuromorphic hardware as applied to the healthcare domain. We benchmark various hardware platforms by performing a sensor fusion signal processing task combining electromyography (EMG) signals with computer vision. Comparisons are made between dedicated neuromorphic processors and embedded AI accelerators in terms of inference latency and energy. Finally, we provide our analysis of the field and share a perspective on the advantages, disadvantages, challenges, and opportunities that various accelerators and neuromorphic processors introduce to healthcare and biomedical domains.