Organ and tissue transplantation to provide a solution to wounds and deficiencies is a drawback that should urgently be addressed, as there are still problems in the process, such as shortage of donors and the danger of immunological rejection. The implementation of artificial organs and tissues indicates a better quality of life in patients, however, a manufacturing technique is needed to enable the development of complicated constructions. The 3D printing technique addresses recent restrictions in the development of organs and tissues, as it makes it possible to integrate cells into biomaterials for regeneration of different biological constructions. This review summarises the most important studies and progress in the development of tissue implants, prostheses and engineering through 3D printing, introduces the various biomaterials used in additive manufacturing, as well as the multiple printing techniques used in biomedical applications.