Bioprinting techniques have been flourishing in the field of biofabrication with pronounced and exponential developments in the past years. Novel biomaterial inks used for the formation of bioinks have been developed, allowing the manufacturing of in vitro models and implants tested preclinically with a certain degree of success. Furthermore, incredible advances in cell biology, namely, in pluripotent stem cells, have also contributed to the latest milestones where more relevant tissues or organ-like constructs with a certain degree of functionality can already be obtained. These incredible strides have been possible with a multitude of multidisciplinary teams around the world, working to make bioprinted tissues and organs more relevant and functional. Yet, there is still a long way to go until these biofabricated constructs will be able to reach the clinics. In this review, we summarize the main bioprinting activities linking them to tissue and organ development and physiology. Most bioprinting approaches focus on mimicking fully matured tissues. Future bioprinting strategies might pursue earlier developmental stages of tissues and organs. The continuous convergence of the experts in the fields of material sciences, cell biology, engineering, and many other disciplines will gradually allow us to overcome the barriers identified on the demanding path toward manufacturing and adoption of tissue and organ replacements.