Additive manufacturing (AM) is the process of printing 3D objects in a layer-by-layer manner. Polymers and their composites are some of the most widely used materials in modern industries and are of great interest in the field of AM due to their vast potential for various applications, especially in the medical, aerospace, and automotive industries. Many studies have been conducted to develop new polymer materials for AM techniques, which include vat photopolymerization, material jetting, powder bed fusion, material extrusion, binder jetting, and sheet lamination. Although several reviews on the development of polymer materials for AM have been published, most of them only focus on a specific application, process, or type of material. Therefore, this article serves to provide a comprehensive review on the progress in polymer material development for AM techniques. It begins with an introduction to different AM techniques, followed by highlighting the progress of their development. Material requirements, notable advances in newly developed materials and their potential applications are discussed in detail and summarized. This review concludes by identifying the major challenges currently encountered in using AM for polymer materials and providing insights into the valuable opportunities it presents, in hopes of spurring further development in this field.