This paper provides an overview of the current and near-future applications of Artificial Intelligence (AI) in Medicine and Health Care and presents a classification according to their ethical and societal aspects, potential benefits and pitfalls, and issues that can be considered controversial and are not deeply discussed in the literature. This work is based on an analysis of the state of the art of research and technology, including existing software, personal monitoring devices, genetic tests and editing tools, personalized digital models, online platforms, augmented reality devices, and surgical and companion robotics. Motivated by our review, we present and describe the notion of 'extended personalized medicine', we then review existing applications of AI in medicine and healthcare and explore the public perception of medical AI systems, and how they show, simultaneously, extraordinary opportunities and drawbacks that even question fundamental medical concepts. Many of these topics coincide with urgent priorities recently defined by the World Health Organization for the coming decade. In addition, we study the transformations of the roles of doctors and patients in an age of ubiquitous information, identify the risk of a division of Medicine into 'fake-based', 'patient-generated', and 'scientifically tailored', and draw the attention of some aspects that need further thorough analysis and public debate.