Three-dimensional (3D) bioprinting is an emerging manufacturing technology that layers living cells and biocompatible natural or synthetic materials to build complex, functional living tissue with the requisite 3D geometries. This technology holds tremendous promise across a plethora of applications as diverse as regenerative medicine, pathophysiological studies, and drug testing. Despite some success demonstrated in early attempts to recreate complex tissue structures, however, the field of bioprinting is very much in its infancy. There are a variety of challenges to building viable, functional, and lasting 3D structures, not the least of which is translation from a research to a clinical setting. In this review, the current translational status of 3D bioprinting is assessed for several major tissue types in the body (skin, bone/cartilage, cardiovascular, central/peripheral nervous systems, skeletal muscle, kidney, and liver), recent breakthroughs and current challenges are highlighted, and future prospects for this exciting research field are discussed. We begin with an overview of the technology itself, followed by a detailed discussion of the current approaches relevant for bioprinting different tissues for regenerative medicine.