The last two centuries saw groundbreaking advances in the field of healthcare: from the invention of the vaccine to organ transplant, and eradication of numerous deadly diseases. Yet, these breakthroughs have only illuminated the role that individual traits and behaviours play in the health state of a person. Continuous patient monitoring and individually-tailored therapies can help in early detection and efficient tackling of health issues. However, even the most developed nations cannot afford proactive personalised healthcare at scale. Mobile computing devices, nowadays equipped with an array of sensors, high-performance computing power, and carried by their owners at all time, promise to revolutionise modern healthcare. These devices can enable continuous patient monitoring, and, with the help of machine learning, can build predictive models of patient's health and behaviour. Finally, through their close integration with a user's lifestyle mobiles can be used to deliver personalised proactive therapies. In this article, we develop the concept of anticipatory mobile-based healthcare - anticipatory mobile digital health - and examine the opportunities and challenges associated with its practical realisation.