Three-dimensional (3D) printing, as one of the most popular recent additive manufacturing processes, has shown strong potential for the fabrication of biostructures in the field of tissue engineering, most notably for bones, orthopedic tissues, and associated organs. Desirable biological, structural, and mechanical properties can be achieved for 3D-printed constructs with a proper selection of biomaterials and compatible bioprinting methods, possibly even while combining additive and conventional manufacturing (AM and CM) procedures. However, challenges remain in the need for improved printing resolution (especially at the nanometer level), speed, and biomaterial compatibilities, and a broader range of suitable 3D-printed materials. This review provides an overview of recent advances in the development of 3D bioprinting techniques, particularly new hybrid 3D bioprinting technologies for combining the strengths of both AM and CM, along with a comprehensive set of material selection principles, promising medical applications, and limitations and future prospects.