Bioprinting is an emerging technology for fabricating intricate and diverse structures that closely mimic natural tissues and organs for such applications as tissue engineering, drug delivery, and cancer research as well. Among the various bioprinting techniques, extrusion-based bioprinting stands out due to its capability to apply a wide range of biomaterials and living cells and its controllability over printed structures. In bioprinting, the bioink stored in a syringe is forced to flow through the nozzle connected to the syringe, and then to exit and deposit onto the printing stage to form three-dimensional (3D) structures. The bioprinting process involves the flow of bioink in both syringe and nozzle and then its flow or spreading on a printing stage. As a result, fluid mechanics plays a crucial role in extrusion bioprinting. Notably, the biomaterials used in bioprinting are typically non-Newtonian fluids, which have complex viscoelastic and thixotropic behaviors; and the influence of these behaviors on the bioprinting process has been drawn considerable attention by employing various methods, including the numerical simulations via computational fluid dynamics (CFD). This paper reviews the latest development in the fluid mechanics aspects of extrusion-based bioprinting to shed light on the challenges and key considerations involved. It covers the topics of extrusion bioprinting (including driving mechanisms, printability, cell viability), biomaterial rheology and its effect on bioprinting, multi-material bioprinting and numerical simulation of bioprinting. Key issues and challenges are also discussed along with the recommendations for future research.