Three-dimensional (3D) printing is well acknowledged to constitute an important technology in tissue engineering, largely due to the increasing global demand for organ replacement and tissue regeneration. In 3D bioprinting, which is a step ahead of 3D biomaterial printing, the ink employed is impregnated with cells, without compromising ink printability. This allows for immediate scaffold cellularization and generation of complex structures. The use of cell-laden inks or bio-inks provides the opportunity for enhanced cell differentiation for organ fabrication and regeneration. Recognizing the importance of such bio-inks, the current study comprehensively explores the state of the art of the utilization of bio-inks based on natural polymers (biopolymers), such as cellulose, agarose, alginate, decellularized matrix, in 3D bioprinting. Discussions regarding progress in bioprinting, techniques and approaches employed in the bioprinting of natural polymers, and limitations and prospects concerning future trends in human-scale tissue and organ fabrication are also presented.