Bioink properties before, during and after 3D bioprinting are crucial for the printability and functionality of bioprinted tissues. Bioinks, which are cytocompatible hydrogel precursors, are used in additive manufacturing to create structures containing living cells. The properties of these bioinks, such as structural resolution, shape fidelity, and cell survival, are essential for the success of bioprinting. The final properties of the bioprinted tissue construct are influenced by the number of cells present, their proliferation, migration, and interaction with the material. A calibrated computational framework can predict tissue development and optimize bioprinting parameters such as the starting material, initial cell loading, and construct geometry. This review discusses the properties of bioinks and their interaction with cells, focusing on the most popular bioprinting approaches. The effect of cells on hydrogel processing and vice versa is highlighted, along with numerical approaches for depicting cellular mechanics and predicting mechanical properties. The review also covers the different bioprinting methodologies, including inkjet, orifice-free, and extrusion bioprinting, and the suitable bioink materials for each. The properties of hydrogels, such as viscosity, shear thinning, and thixotropy, are important for the printing process and cell survival. The gelation methods, including physical, chemical, and photo-crosslinking, are used to ensure the stability of bioprinted constructs. The review also discusses the commercialization of bioprinting, the availability of commercially available bioprinters and bioinks, and the mechanical properties of hydrogels. The effect of cell content on material processing is also discussed, as well as the prediction of properties of hydrogels containing living cells. The review concludes with future perspectives on bioprinting, including the potential for tissue engineering applications and the need for further research in this field.Bioink properties before, during and after 3D bioprinting are crucial for the printability and functionality of bioprinted tissues. Bioinks, which are cytocompatible hydrogel precursors, are used in additive manufacturing to create structures containing living cells. The properties of these bioinks, such as structural resolution, shape fidelity, and cell survival, are essential for the success of bioprinting. The final properties of the bioprinted tissue construct are influenced by the number of cells present, their proliferation, migration, and interaction with the material. A calibrated computational framework can predict tissue development and optimize bioprinting parameters such as the starting material, initial cell loading, and construct geometry. This review discusses the properties of bioinks and their interaction with cells, focusing on the most popular bioprinting approaches. The effect of cells on hydrogel processing and vice versa is highlighted, along with numerical approaches for depicting cellular mechanics and predicting mechanical properties. The review also covers the different bioprinting methodologies, including inkjet, orifice-free, and extrusion bioprinting, and the suitable bioink materials for each. The properties of hydrogels, such as viscosity, shear thinning, and thixotropy, are important for the printing process and cell survival. The gelation methods, including physical, chemical, and photo-crosslinking, are used to ensure the stability of bioprinted constructs. The review also discusses the commercialization of bioprinting, the availability of commercially available bioprinters and bioinks, and the mechanical properties of hydrogels. The effect of cell content on material processing is also discussed, as well as the prediction of properties of hydrogels containing living cells. The review concludes with future perspectives on bioprinting, including the potential for tissue engineering applications and the need for further research in this field.