Regenerative medicine holds the promise of engineering functional tissues or organs to heal or replace abnormal and necrotic tissues/organs, offering hope for filling the gap between organ shortage and transplantation needs. Three-dimensional (3D) bioprinting is evolving into an unparalleled biomanufacturing technology due to its high-integration potential for patient-specific designs, precise and rapid manufacturing capabilities with high resolution, and unprecedented versatility. It enables precise control over multiple compositions, spatial distributions, and architectural accuracy/complexity, therefore achieving effective recapitulation of microstructure, architecture, mechanical properties, and biological functions of target tissues and organs. Here we provide an overview of recent advances in 3D bioprinting technology, as well as design concepts of bioinks suitable for the bioprinting process. We focus on the applications of this technology for engineering living organs, focusing more specifically on vasculature, neural networks, the heart and liver. We conclude with current challenges and the technical perspective for further development of 3D organ bioprinting.