A digital twin is a virtual model of a physical entity, with dynamic, bi-directional links between the physical entity and its corresponding twin in the digital domain. Digital twins are increasingly used today in different industry sectors. Applied to medicine and public health, digital twin technology can drive a much-needed radical transformation of traditional electronic health/medical records (focusing on individuals) and their aggregates (covering populations) to make them ready for a new era of precision (and accuracy) medicine and public health. Digital twins enable learning and discovering new knowledge, new hypothesis generation and testing, and experiments and comparisons. They are poised to play a key role in formulating highly personalised treatments and interventions in the future. This paper provides an overview of the technology's history and main concepts. A number of application examples of digital twins for personalised medicine, public health, and smart healthy cities are presented, followed by a brief discussion of the key technical and other challenges involved in such applications, including ethical issues that arise when digital twins are applied to model humans.