Three-dimensional (3D) organ bioprinting is an attractive scientific area with huge commercial profit, which could solve all the serious bottleneck problems for allograft transplantation, high-throughput drug screening, and pathological analysis. Integrating multiple heterogeneous adult cell types and/or stem cells along with other biomaterials (e.g., polymers, bioactive agents, or biomolecules) to make 3D constructs functional is one of the core issues for 3D bioprinting of bioartificial organs. Both natural and synthetic polymers play essential and ubiquitous roles for hierarchical vascular and neural network formation in 3D printed constructs based on their specific physical, chemical, biological, and physiological properties. In this article, several advanced polymers with excellent biocompatibility, biodegradability, 3D printability, and structural stability are reviewed. The challenges and perspectives of polymers for rapid manufacturing of complex organs, such as the liver, heart, kidney, lung, breast, and brain, are outlined.