The development of artificial tissue/organs with the functional maturity of their native equivalents is one of the long-awaited panaceas for the medical and pharmaceutical industries. Advanced 3D cell-printing technology and various functional bioinks are promising technologies in the field of tissue engineering that have enabled the fabrication of complex 3D living tissue/organs. Various requirements for these tissues, including a complex and large-volume structure, tissue-specific microenvironments, and functional vasculatures, have been addressed to develop engineered tissue/organs with native relevance. Functional tissue/organ constructs have been developed that satisfy such criteria and may facilitate both in vivo replenishment of damaged tissue and the development of reliable in vitro testing platforms for drug development. This review describes key developments in technologies and materials for engineering 3D cell-printed constructs for therapeutic and drug testing applications.