Bioprinting, the three-dimensional (3D) printing of cell-laden inks, will be a truly revolutionary technology for the biomaterials community. The number of bioink studies, especially aimed at functional tissues, remains significantly limited, and furthermore, current bioinks are limited by a narrow window of printability. This can be largely attributed to the fact that the preparation of bioinks and their 3D printing is significantly complicated by the presence of cells, which require strict conditions for their viability. This article discusses how cells should be considered during bioink synthesis, 3D printing, and post-printing processing. We also discuss what has been reported thus far with regard to the relationships between bioink material properties and cells. This underlines the need for next-generation bioinks that simultaneously achieve excellent printability, high cell viability, and a wide range of material properties.