The use of three-dimensional bioprinting technology combined with the principle of tissue engineering is important for the construction of tissue or organ regeneration microenvironments. As a three-dimensional bioprinting ink, hydrogels need to be highly printable and provide a stiff and cell-friendly microenvironment. At present, hydrogels are used as bioprinting inks in tissue engineering. However, there is still a lack of summary of the latest 3D printing technology and the properties of hydrogel materials. In this paper, the materials commonly used as hydrogel bioinks; the advanced technologies including inkjet bioprinting, extrusion bioprinting, laser-assisted bioprinting, stereolithography bioprinting, suspension bioprinting, and digital 3D bioprinting technologies; printing characterization including printability and fidelity; biological properties, and the application fields of bioprinting hydrogels in bone tissue engineering, skin tissue engineering, cardiovascular tissue engineering are reviewed, and the current problems and future directions are prospected.