The introduction of sensor technology in our daily lives has brought comfort, convenience, and improved health over the past few decades. Technological advances further expanded the use of medical sensors by reducing their size and costs. Medical sensors improve the intelligence and capabilities of healthcare services including, remote health monitoring, surgical procedures, therapy, and rehabilitation. We present a comprehensive review of medical sensors in the last 50 years focusing on their deployment in healthcare applications. The review also discusses the role of Internet of Things (IoT) technology in enhancing the capabilities of sensor technologies for the healthcare domain. Moreover, we also investigate the benefits and challenges of various integrated architectures which have been proposed recently to seamlessly integrate heterogeneous medical sensors with emerging technologies and paradigms that include edge, Mobile Cloud Computing (MCC), fog, and cloud computing technologies. Finally, we identify future challenges that must be addressed to achieve the maximum potential and benefits of medical sensor technologies and ultimately provide robust, scalable, reliable, and cost-effective healthcare delivery.