Although healthcare is a remarkably sensitive domain of application, and systems that exert direct control over the world can cause harm in a way that humans cannot necessarily correct or oversee, it is still unclear whether and how healthcare robots are currently regulated or should be regulated. Existing regulations are primarily unprepared to provide guidance for such a rapidly evolving field and accommodate devices that rely on machine learning and AI. Moreover, the field of healthcare robotics is very rich and extensive, but it is still very much scattered and unclear in terms of definitions, medical and technical classifications, product characteristics, purpose, and intended use. As a result, these devices often navigate between the medical device regulation or other non-medical norms, such as the ISO personal care standard. Before regulating the field of healthcare robots, it is therefore essential to map the major state-of-the-art developments in healthcare robotics, their capabilities and applications, and the challenges we face as a result of their integration within the healthcare environment. This contribution fills in this gap and lack of clarity currently experienced within healthcare robotics and its governance by providing a structured overview of and further elaboration on the main categories now established, their intended purpose, use, and main characteristics. We explicitly focus on surgical, assistive, and service robots to rightfully match the definition of healthcare as the organized provision of medical care to individuals, including efforts to maintain, treat, or restore physical, mental, or emotional well-being. We complement these findings with policy recommendations to help policymakers unravel an optimal regulatory framing for healthcare robot technologies