The tooth and its supporting tissues are organized with complex three-dimensional (3D) architecture, including the dental pulp with a blood supply and nerve tissues, complex multilayer periodontium, and highly aligned periodontal ligament (PDL). Mimicking such 3D complexity and the multicellular interactions naturally existing in dental structures represents great challenges in dental regeneration. Attempts to construct the complex system of the tooth and tooth-supporting apparatus (i.e., the PDL, alveolar bone, and cementum) have made certain progress owing to 3D printing biotechnology. Recent advances have enabled the 3D printing of biocompatible materials, seed cells, and supporting components into complex 3D functional living tissue. Furthermore, 3D bioprinting is driving major innovations in regenerative medicine, giving the field of regenerative dentistry a boost. The fabrication of scaffolds via 3D printing is already being performed extensively at the laboratory bench and in clinical trials; however, printing living cells and matrix materials together to produce tissue constructs by 3D bioprinting remains limited to the regeneration of dental pulp and the tooth germ. This review summarizes the application of scaffolds for cell seeding and biofabricated tissues via 3D printing and bioprinting, respectively, in the tooth and its supporting tissues. Additionally, the key advantages and prospects of 3D bioprinting in regenerative dentistry are highlighted, providing new ideas for dental regeneration.