Physical organ models are the objects that replicate the patient-specific anatomy and have played important roles in modern medical diagnosis and disease treatment. 3D printing, as a powerful multi-function manufacturing technology, breaks the limitations of traditional methods and provides a great potential for manufacturing organ models. However, the clinical application of organ model is still in small scale, facing the challenges including high cost, poor mimicking performance and insufficient accuracy. In this review, the mainstream 3D printing technologies are introduced, and the existing manufacturing methods are divided into "directly printing" and "indirectly printing", with an emphasis on choosing suitable techniques and materials. This review also summarizes the ideas to address these challenges and focuses on three points: 1) what are the characteristics and requirements of organ models in different application scenarios, 2) how to choose the suitable 3D printing methods and materials according to different application categories, and 3) how to reduce the cost of organ models and make the process simple and convenient. Moreover, the state-of-the-art in organ models are summarized and the contribution of 3D printed organ models to various surgical procedures is highlighted. Finally, current limitations, evaluation criteria and future perspectives for this emerging area are discussed.