Large-scale Artificial General Intelligence (AGI) models, including Large Language Models (LLMs) such as ChatGPT/GPT-4, have achieved unprecedented success in a variety of general domain tasks. Yet, when applied directly to specialized domains like medical imaging, which require in-depth expertise, these models face notable challenges arising from the medical field's inherent complexities and unique characteristics. In this review, we delve into the potential applications of AGI models in medical imaging and healthcare, with a primary focus on LLMs, Large Vision Models, and Large Multimodal Models. We provide a thorough overview of the key features and enabling techniques of LLMs and AGI, and further examine the roadmaps guiding the evolution and implementation of AGI models in the medical sector, summarizing their present applications, potentialities, and associated challenges. In addition, we highlight potential future research directions, offering a holistic view on upcoming ventures. This comprehensive review aims to offer insights into the future implications of AGI in medical imaging, healthcare, and beyond.