In recent years, there has been a growing demand for miniaturization, low power consumption, quick treatments, and non-invasive clinical strategies in the healthcare industry. To meet these demands, healthcare professionals are seeking new technological paradigms that can improve diagnostic accuracy while ensuring patient compliance. Neuromorphic engineering, which uses neural models in hardware and software to replicate brain-like behaviors, can help usher in a new era of medicine by delivering low power, low latency, small footprint, and high bandwidth solutions. This paper provides an overview of recent neuromorphic advancements in medicine, including medical imaging and cancer diagnosis, processing of biosignals for diagnosis, and biomedical interfaces, such as motor, cognitive, and perception prostheses. For each section, we provide examples of how brain-inspired models can successfully compete with conventional artificial intelligence algorithms, demonstrating the potential of neuromorphic engineering to meet demands and improve patient outcomes. Lastly, we discuss current struggles in fitting neuromorphic hardware with non-neuromorphic technologies and propose potential solutions for future bottlenecks in hardware compatibility.