Extrusion-based bioprinting involves extrusion of bioinks through nozzles to create three-dimensional structures. The bioink contains living organisms with biological relevance for emerging applications such as tissue scaffolds, organs-on-a-chip, regenerative medicine, and drug delivery systems. Bioinks, which are mixtures of biomaterials and living cells, influence the quality of printed constructs through their physical, mechanical, biological, and rheological behavior. Printability is a property of a bioink used to describe its ability to create well-defined structures. Amongst all contributing factors, rheological properties and printing parameters are primary factors that influence the quality of bioprinted constructs. With the increasing popularity of extrusion bioprinting, different approaches for controlling these properties and parameters have emerged. This review highlights the role of rheology and process parameters in extrusion bioprinting and discusses qualitative and quantitative methods proposed to measure and define the printability of bioinks. Finally, an overview of key challenges and future trends in extrusion bioprinting is provided.