Bioprinting offers a highly-automated and advanced manufacturing platform that facilitates the deposition of bio-inks (living cells, biomaterials and growth factors) in a scalable and reproducible manner, a process that is lacking in conventional tissue engineering approaches. Significant improvements in the field of bioprinting have occurred over the last two decades. This reviews provides an in-depth analysis of recent improvements in the bioprinting techniques, progress in bio-ink development, implementation of new bioprinting and tissue maturation strategies. Special attention is givent to the role of polymer science and how it complements 3D bioprinting to overcome some of the major impediments in the field of organ printing. A concise overview of the anatomy and physiology of different tissues/organs is provided, followed by important design considerations to better facilitate the fabrication of biomimetic tissues/organs for tissue engineering and regenerative medicine (TERM). Last, a realistic overview of current status in organ bioprinting is presented, including recent accomplishments in bioprinting tissue-engineered constructs, the limitations and challenges, as well as opportunities for future research. We strongly believe that with the advances in polymer sciences, it will be an impending reality for on-demand bioprinting of patient-specific tissues/organs.