Tissue engineering for injured tissue replacement and regeneration has been a subject of investigation over the last 30 years, and there has been considerable interest in using additive manufacturing to achieve these goals. Despite such efforts, many key questions remain unanswered, particularly in the area of biomaterial selection for these applications as well as quantitative understanding of the process science. The strategic utilization of biological macromolecules provides a versatile approach to meet diverse requirements in 3D printing, such as printability, buildability, and biocompatibility. These molecules play a pivotal role in both physical and chemical cross-linking processes throughout the biofabrication, contributing significantly to the overall success of the 3D printing process. Among the several bioprintable materials, gelatin methacryloyl (GelMA) has been widely utilized for diverse tissue engineering applications, with some degree of success. In this context, this review will discuss the key bioengineering approaches to identify the gelation and cross-linking strategies that are appropriate to control the rheology, printability, and buildability of biomaterial inks. This review will focus on the GelMA as the structural (scaffold) biomaterial for different tissues and as a potential carrier vehicle for the transport of living cells as well as their maintenance and viability in the physiological system. Recognizing the importance of printability toward shape fidelity and biophysical properties, a major focus in this review has been to discuss the qualitative and quantitative impact of the key factors, including microrheological, viscoelastic, gelation, shear thinning properties of biomaterial inks, and printing parameters, in particular, reference to 3D extrusion printing of GelMA-based biomaterial inks. Specifically, we emphasize the different possibilities to regulate mechanical, swelling, biodegradation, and cellular functionalities of GelMA-based bio(material) inks, by hybridization techniques, including different synthetic and natural biopolymers, inorganic nanofillers, and microcarriers. At the close, the potential possibility of the integration of experimental data sets and artificial intelligence/machine learning approaches is emphasized to predict the printability, shape fidelity, or biophysical properties of GelMA bio(material) inks for clinically relevant tissues.