Bone is the second most commonly transplanted tissue in the world, with at least four million surgical procedures using bone grafts and bone substitutes every year. However, the limitations of traditional treatments have affected current treatment options, and the clinical demand for bone grafts has continued to increase due to the high incidence of trauma, cancer, infection, and arthritis. Autografts and allografts are commonly used in the clinical treatment of bone defects, but chronic inflammation, disease transmission, and immune rejection have hindered their development. In addition, metal-based material scaffolds are the most widely used implants, while there are also problems such as stress shield, infection, and inflammation, leading researchers to find new material scaffolds to replace them. Therefore, developing bioactive three-dimensional (3D) scaffolds that can be adaptively expanded and filled to promote bone regeneration has become a key area of focus in bone tissue engineering (BTE). In recent years, manufacturing methods including 4D printing of shape memory materials have been used to create new methods to replace traditional bone grafts. This paper mainly reviews the classification of traditional polymer materials and new shape memory polymer materials, the main manufacturing methods, mechanical properties, biocompatibility, and the latest applications of polymer scaffolds in BTE. Furthermore, the importance, current challenges, and future development directions of 4D printing technology in BTE are summarized.