Tissue engineering is still far from the set goal, production of functional replacement human organs and it is reasonable to assume that biofabrication technology will probably not allow the creation of 100% authentic copies of functional organs. A more realistic technological goal is to design tissue and organ constructs with the ability to restore function of damaged organs. Thanks to the fast-evolving biomimetic approach which represents the integration of engineering and developmental biology, that goal has become feasible. The rapid progress in this area was especially stimulated by the beginning of use of 3D printers in the development of biological structures or bioprinting. This paper describes the traditional approaches to tissue engineering, highlights their shortcomings, compares them with newer methods of biomimetic approach and presents recent advances in the development of the living structures which were enabled by the revolutionary technology of bioprinting.