Foundation models (FMs) are large-scale deep learning models trained on massive datasets, often using self-supervised learning techniques. These models serve as a versatile base for a wide range of downstream tasks, including those in medicine and healthcare. FMs have demonstrated remarkable success across multiple healthcare domains. However, existing surveys in this field do not comprehensively cover all areas where FMs have made significant strides. In this survey, we present a comprehensive review of FMs in medicine, focusing on their evolution, learning strategies, flagship models, applications, and associated challenges. We examine how prominent FMs, such as the BERT and GPT families, are transforming various aspects of healthcare, including clinical large language models, medical image analysis, and omics research. Additionally, we provide a detailed taxonomy of FM-enabled healthcare applications, spanning clinical natural language processing, medical computer vision, graph learning, and other biology- and omics- related tasks. Despite the transformative potentials of FMs, they also pose unique challenges. This survey delves into these challenges and highlights open research questions and lessons learned to guide researchers and practitioners. Our goal is to provide valuable insights into the capabilities of FMs in health, facilitating responsible deployment and mitigating associated risks.