Bioprinting is a process based on additive manufacturing from materials containing living cells. These materials, often referred to as bioink, are based on cytocompatible hydrogel precursor formulations, which gel in a manner compatible with different bioprinting approaches. The bioink properties before, during and after gelation are essential for its printability, comprising such features as achievable structural resolution, shape fidelity and cell survival. However, it is the final properties of the matured bioprinted tissue construct that are crucial for the end application. During tissue formation these properties are influenced by the amount of cells present in the construct, their proliferation, migration and interaction with the material. A calibrated computational framework is able to predict the tissue development and maturation and to optimize the bioprinting input parameters such as the starting material, the initial cell loading and the construct geometry. In this contribution relevant bioink properties are reviewed and discussed on the example of most popular bioprinting approaches. The effect of cells on hydrogel processing and vice versa is highlighted. Furthermore, numerical approaches were reviewed and implemented for depicting the cellular mechanics within the hydrogel as well as for prediction of mechanical properties to achieve the desired hydrogel construct considering cell density, distribution and material-cell interaction.