Three-dimensional (3D) bioprinting, as a freeform biomedical manufacturing approach, has been increasingly adopted for the fabrication of constructs analogous to living tissues. Generally, materials printed during 3D bioprinting are referred as bioinks, which may include living cells, extracellular matrix materials, cell media, and/or other additives. For 3D bioprinting to be an enabling tissue engineering approach, the bioink printability is a critical requirement as tissue constructs must be able to be printed and reproduce the complex micro-architecture of native tissues in vitro in sufficient resolution. The bioink printability is generally characterized in terms of the controllable formation of well-defined droplets/jets/filaments and/or the morphology and shape fidelity of deposited building blocks. This review presents a comprehensive overview of the studies of bioink printability during representative 3D bioprinting processes, including inkjet printing, laser printing, and micro-extrusion, with a focus on the understanding of the underlying physics during the formation of bioink-based features. A detailed discussion is conducted based on the typical time scales and dimensionless quantities for printability evaluation during bioprinting. For inkjet printing, the Z (the inverse of the Ohnesorge number), Weber, and capillary numbers have been employed for the construction of phase diagrams during the printing of Newtonian fluids, while the Weissenberg and Deborah numbers have been utilized during the printing of non-Newtonian bioinks. During laser printing of Newtonian solutions, the jettability can be characterized using the inverse of the Ohnesorge number, while Ohnesorge, elasto-capillary, and Weber numbers have been utilized to construct phase diagrams for typical non-Newtonian bioinks. For micro-extrusion, seven filament types have been identified including three types of well-defined filaments and four types of irregular filaments. During micro-extrusion, the Oldroyd number has been used to characterize the dimensions of the yielded areas of Herschel-Bulkley fluids. Non-ideal jetting behaviors are common during the droplet-based inkjet and laser printing processes due to the local nonuniformity and nonhomogeneity of cell-laden bioinks.