Various cell aggregate culture technologies have been developed and actively applied to tissue engineering and organ-on-a-chip. However, the conventional culture technologies are labor-intensive, and their outcomes are highly user dependent. In addition, the technologies cannot be used to produce three-dimensional (3D) complex tissues. In this regard, 3D cell aggregate printing technology has attracted increased attention from many researchers owing to its 3D processability. The technology allows the fabrication of 3D freeform constructs using multiple types of cell aggregates in an automated manner. Technological advancement has resulted in the development of a printing technology with a high resolution of approximately 20 μm in 3D space. A high-speed printing technology that can print a cell aggregate in milliseconds has also been introduced. The developed aggregate printing technologies are being actively applied to produce various types of engineered tissues. Although various types of high-performance printing technologies have been developed, there are still some technical obstacles in the fabrication of engineered tissues that mimic the structure and function of native tissues. This review highlights the central importance and current technical level of 3D cell aggregate printing technology, and their applications to tissue/disease models, artificial tissues, and drug-screening platforms. The paper also discusses the remaining hurdles and future directions of the printing processes.