The field of regenerative medicine has progressed tremendously over the past few decades in its ability to fabricate functional tissue substitutes. Conventional approaches based on scaffolding and microengineering are limited in their capacity of producing tissue constructs with precise biomimetic properties. Three-dimensional (3D) bioprinting technology, on the other hand, promises to bridge the divergence between artificially engineered tissue constructs and native tissues. In a sense, 3D bioprinting offers unprecedented versatility to co-deliver cells and biomaterials with precise control over their compositions, spatial distributions, and architectural accuracy, therefore achieving detailed or even personalized recapitulation of the fine shape, structure, and architecture of target tissues and organs. Here we briefly describe recent progresses of 3D bioprinting technology and associated bioinks suitable for the printing process. We then focus on the applications of this technology in fabrication of biomimetic constructs of several representative tissues and organs, including blood vessel, heart, liver, and cartilage. We finally conclude with future challenges in 3D bioprinting as well as potential solutions for further development.