In recent years, with the current advancements in Robotics and Artificial Intelligence (AI), robots have the potential to support the field of healthcare. Robotic systems are often introduced in the care of the elderly, children, and persons with disabilities, in hospitals, in rehabilitation and walking assistance, and other healthcare situations. In this survey paper, the recent advances in robotic technology applied in the healthcare domain are discussed. The paper provides detailed information about state-of-the-art research in care, hospital, assistive, rehabilitation, and walking assisting robots. The paper also discusses the open challenges healthcare robots face to be integrated into our society.