Over the past several decades, there has been an ever-increasing demand for organ transplants. However, there is a severe shortage of donor organs, and as a result of the increasing demand, the gap between supply and demand continues to widen. A potential solution to this problem is to grow or fabricate organs using biomaterial scaffolds and a person's own cells. Although the realization of this solution has been limited, the development of new biofabrication approaches has made it more realistic. This review provides an overview of natural and synthetic biomaterials that have been used for organ/tissue development. It then discusses past and current biofabrication techniques, with a brief explanation of the state of the art. Finally, the review highlights the need for combining vascularization strategies with current biofabrication techniques. Given the multitude of applications of biofabrication technologies, from organ/tissue development to drug discovery/screening to development of complex in vitro models of human diseases, these manufacturing technologies can have a significant impact on the future of medicine and health care.