Three-dimensional (3D) bioprinting is on the cusp of permitting the direct fabrication of artificial living tissue. Multicellular building blocks (bioinks) are dispensed layer by layer and scaled for the target construct. However, only a few materials are able to fulfill the considerable requirements for suitable bioink formulation, a critical component of efficient 3D bioprinting. Alginate, a naturally occurring polysaccharide, is clearly the most commonly employed material in current bioinks. Here, we discuss the benefits and disadvantages of the use of alginate in 3D bioprinting by summarizing the most recent studies that used alginate for printing vascular tissue, bone and cartilage. In addition, other breakthroughs in the use of alginate in bioprinting are discussed, including strategies to improve its structural and degradation characteristics. In this review, we organize the available literature in order to inspire and accelerate novel alginate-based bioink formulations with enhanced properties for future applications in basic research, drug screening and regenerative medicine.