There is a growing demand for alternative fabrication approaches to develop tissues and organs as conventional techniques are not capable of fabricating constructs with required structural, mechanical, and biological complexity. 3D bioprinting offers great potential to fabricate highly complex constructs with precise control of structure, mechanics, and biological matter [i.e., cells and extracellular matrix (ECM) components]. 3D bioprinting is an additive manufacturing approach that utilizes a “bioink” to fabricate devices and scaffolds in a layer-by-layer manner. 3D bioprinting allows printing of a cell suspension into a tissue construct with or without a scaffold support. The most common bioinks are cell-laden hydrogels, decellulerized ECM-based solutions, and cell suspensions. In this mini review, a brief description and comparison of the bioprinting methods, including extrusion-based, droplet-based, and laser-based bioprinting, with particular focus on bioink design requirements are presented. We also present the current state of the art in bioink design including the challenges and future directions.