Physics-inspired generative models (GMs), in particular diffusion models and Poisson flow models, enhance Bayesian methods and promise great utility in medical imaging. This review examines the transformative role of such generative methods. First, a variety of physics-inspired GMs, including denoising diffusion probabilistic models, score-based diffusion models, and Poisson flow generative models (including PFGM++), are revisited, with an emphasis on their accuracy, robustness and acceleration. Then, major applications of physics-inspired GMs in medical imaging are presented, comprising image reconstruction, image generation, and image analysis. Finally, future research directions are brainstormed, including unification of physics-inspired GMs, integration with vision-language models, and potential novel applications of GMs. Since the development of generative methods has been rapid, it is hoped that this review will give peers and learners a timely snapshot of this new family of physics-driven GMs and help capitalize their enormous potential for medical imaging.