The scope of three-dimensional printing is expanding rapidly, with innovative approaches resulting in the evolution of state-of-the-art 3D bioprinting (3DbioP) techniques for solving issues in bioengineering and biopharmaceutical research. The methods and tools in 3DbioP emphasize the extrusion process, bioink formulation, and stability of the bioprinted scaffold. Thus, 3DbioP technology augments 3DP in the biological world by providing technical support to regenerative therapy, drug delivery, bioengineering of prosthetics, and drug kinetics research. Besides the above, drug delivery and dosage control have been achieved using 3D bioprinted microcarriers and capsules. Developing a stable, biocompatible, and versatile bioink is a primary requisite in biofabrication. The 3DbioP research is breaking the technical barriers at a breakneck speed. Numerous techniques and biomaterial advancements have helped to overcome current 3DbioP issues related to printability, stability, and bioink formulation. Therefore, this Review aims to provide an insight into the technical challenges of bioprinting, novel biomaterials for bioink formulation, and recently developed 3D bioprinting methods driving future applications in biofabrication research.