Large AI models, or foundation models, are models recently emerging with massive scales both parameter-wise and data-wise, the magnitudes of which can reach beyond billions. Once pretrained, large AI models demonstrate impressive performance in various downstream tasks. A prime example is ChatGPT, whose capability has compelled people's imagination about the far-reaching influence that large AI models can have and their potential to transform different domains of our lives. In health informatics, the advent of large AI models has brought new paradigms for the design of methodologies. The scale of multi-modal data in the biomedical and health domain has been ever-expanding especially since the community embraced the era of deep learning, which provides the ground to develop, validate, and advance large AI models for breakthroughs in health-related areas. This article presents a comprehensive review of large AI models, from background to their applications. We identify seven key sectors in which large AI models are applicable and might have substantial influence, including: 1) bioinformatics; 2) medical diagnosis; 3) medical imaging; 4) medical informatics; 5) medical education; 6) public health; and 7) medical robotics. We examine their challenges, followed by a critical discussion about potential future directions and pitfalls of large AI models in transforming the field of health informatics.