Bioprinting researchers agree that "printability" is a key characteristic for bioink development, but neither the meaning of the term nor the best way to experimentally measure it has been established. Furthermore, little is known with respect to the underlying mechanisms which determine a bioink's printability. A thorough understanding of these mechanisms is key to the intentional design of new bioinks. For the purposes of this review, the domain of printability is defined as the bioink requirements which are unique to bioprinting and occur during the printing process. Within this domain, the different aspects of printability and the factors which influence them are reviewed. The extrudability, filament classification, shape fidelity, and printing accuracy of bioinks are examined in detail with respect to their rheological properties, chemical structure, and printing parameters. These relationships are discussed and areas where further research is needed, are identified. This review serves to aid the bioink development process, which will continue to play a major role in the successes and failures of bioprinting, tissue engineering, and regenerative medicine going forward.