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The application of 3D printing technologies fields for biological tissues, organs, and cells in the context of medical and biotechnology applications requires a significant amount of innovation in a narrow printability range. 3D bioprinting is one such way of addressing critical design challenges in tissue engineering. In a more general sense, 3D printing has become essential in customized implant designing, faithful reproduction of microenvironmental niches, sustainable development of implants, in the capacity to address issues of effective cellular integration, and long-term stability of the cellular constructs in tissue engineering. This review covers various aspects of 3D bioprinting, describes the current state-of-the-art solutions for all aforementioned critical issues, and includes various illustrative representations of technologies supporting the development of phases of 3D bioprinting. It also demonstrates several bio-inks and their properties crucial for being used for 3D printing applications. The review focus on bringing together different examples and current trends in tissue engineering applications, including bone, cartilage, muscles, neuron, skin, esophagus, trachea, tympanic membrane, cornea, blood vessel, immune system, and tumor models utilizing 3D printing technology and to provide an outlook of the future potentials and barriers.