In this article, we review the extant social science and ethical literature on three-dimensional (3D) bioprinting. 3D bioprinting has the potential to be a ‘game-changer’, printing human organs on demand, no longer necessitating the need for living or deceased human donation or animal transplantation. Although the technology is not yet at the level required to bioprint an entire organ, 3D bioprinting may have a variety of other mid-term and short-term benefits that also have positive ethical consequences, for example, creating alternatives to animal testing, filling a therapeutic need for minors and avoiding species boundary crossing. Despite a lack of current socioethical engagement with the consequences of the technology, we outline what we see as some preliminary practical, ethical and regulatory issues that need tackling. These relate to managing public expectations and the continuing reliance on technoscientific solutions to diseases that affect high-income countries. Avoiding prescribing a course of action for the way forward in terms of research agendas, we do briefly outline one possible ethical framework ‘Responsible Research Innovation’ as an oversight model should 3D bioprinting promises are ever realised. 3D bioprinting has a lot to offer in the course of time should it move beyond a conceptual therapy, but is an area that requires ethical oversight and regulation and debate, in the here and now. The purpose of this article is to begin that discussion.