Recent advances in bioprinting technology have been used to precisely dispense cell-laden biomaterials for the construction of complex 3D functional living tissues or artificial organs. Organ printing and biofabrication provides great potential for the freeform fabrication of 3D living organs using cellular spheroids, biocomposite nanofibers, or bioinks as building blocks for regenerative therapy. Vascularization is often identified as a main technological barrier for building 3D organs in tissue engineering. 3D printing of living tissues starts with potential support of biomaterials to maintain structural integrity and degradation of certain time periods after printing of the scaffolds. Biofabrication is the production of complex living and nonliving biological products from raw materials such as cells, molecules, ECM, and biomaterials. Generally, two basic methods are used for the fabrication of scaffolds such as conventional/traditional fabrication processes and advance fabrication processes for engineering organs. A wide range of polymers and biomaterials are used for the fabrication of scaffolds in tissue engineering applications. 3D additive manufacturing is advancing day-by-day; however, there are various critical challenging factors used for fabricating 3D scaffolds. This review is aimed at understanding the various scaffold fabrication techniques, types of polymers and biomaterials used for the fabrication processes, various fields of applications, and different challenges faced in their fabrication of scaffolds in regenerative therapy.