3D bioprinting is a pioneering technology that enables fabrication of biomimetic, multiscale, multi-cellular tissues with highly complex tissue microenvironment, intricate cytoarchitecture, structure-function hierarchy, and tissue-specific compositional and mechanical heterogeneity. Given the huge demand for organ transplantation, coupled with limited organ donors, bioprinting is a potential technology that could solve this crisis of organ shortage by fabrication of fully-functional whole organs. Though organ bioprinting is a far-fetched goal, there has been a considerable and commendable progress in the field of bioprinting that could be used as transplantable tissues in regenerative medicine. This paper presents a first-time review of 3D bioprinting in regenerative medicine, where the current status and contemporary issues of 3D bioprinting pertaining to the eleven organ systems of the human body including skeletal, muscular, nervous, lymphatic, endocrine, reproductive, integumentary, respiratory, digestive, urinary, and circulatory systems were critically reviewed. The implications of 3D bioprinting in drug discovery, development, and delivery systems are also briefly discussed, in terms of in vitro drug testing models, and personalized medicine. While there is a substantial progress in the field of bioprinting in the recent past, there is still a long way to go to fully realize the translational potential of this technology. Computational studies for study of tissue growth or tissue fusion post-printing, improving the scalability of this technology to fabricate human-scale tissues, development of hybrid systems with integration of different bioprinting modalities, formulation of new bioinks with tuneable mechanical and rheological properties, mechanobiological studies on cell-bioink interaction, 4D bioprinting with smart (stimuli-responsive) hydrogels, and addressing the ethical, social, and regulatory issues concerning bioprinting are potential futuristic focus areas that would aid in successful clinical translation of this technology.