Three-dimensional (3D) bioprinting is an additive manufacturing process that utilizes various biomaterials that either contain or interact with living cells and biological systems with the goal of fabricating functional tissue or organ mimics, which will be referred to as bioinks. These bioinks are typically hydrogel-based hybrid systems with many specific features and requirements. The characterizing and fine tuning of bioink properties before, during, and after printing are therefore essential in developing reproducible and stable bioprinted constructs. To date, myriad computational methods, mechanical testing, and rheological evaluations have been used to predict, measure, and optimize bioinks properties and their printability, but none are properly standardized. There is a lack of robust universal guidelines in the field for the evaluation and quantification of bioprintability. In this review, we introduced the concept of bioprintability and discussed the significant roles of various physiomechanical and biological processes in bioprinting fidelity. Furthermore, different quantitative and qualitative methodologies used to assess bioprintability will be reviewed, with a focus on the processes related to pre, during, and post printing. Establishing fully characterized, functional bioink solutions would be a big step towards the effective clinical applications of bioprinted products.