Over the last decade, efforts have been oriented toward the development of suitable gels for 3D printing, with controlled morphology and shear-thinning behavior in well-defined conditions. As a multidisciplinary approach to the fabrication of complex biomaterials, 3D bioprinting combines cells and biocompatible materials, which are subsequently printed in specific shapes to generate 3D structures for regenerative medicine or tissue engineering. A major interest is devoted to the printing of biomimetic materials with structural fidelity after their fabrication. Among some requirements imposed for bioinks, such as biocompatibility, nontoxicity, and the possibility to be sterilized, the nondamaging processability represents a critical issue for the stability and functioning of the 3D constructs. The major challenges in the field of printable gels are to mimic at different length scales the structures existing in nature and to reproduce the functions of the biological systems. Thus, a careful investigation of the rheological characteristics allows a fine-tuning of the material properties that are manufactured for targeted applications. The fluid-like or solid-like behavior of materials in conditions similar to those encountered in additive manufacturing can be monitored through the viscoelastic parameters determined in different shear conditions. The network strength, shear-thinning, yield point, and thixotropy govern bioprintability. An assessment of these rheological features provides significant insights for the design and characterization of printable gels. This review focuses on the rheological properties of printable bioinspired gels as a survey of cutting-edge research toward developing printed materials for additive manufacturing.