Currently, tissue engineering has been dedicated to the development of 3D structures through bioprinting techniques that aim to obtain personalized, dynamic, and complex hydrogel 3D structures. Among the different materials used for the fabrication of such structures, proteins and polysaccharides are the main biological compounds (biopolymers) selected for the bioink formulation. These biomaterials obtained from natural sources are commonly compatible with tissues and cells (biocompatibility), friendly with biological digestion processes (biodegradability), and provide specific macromolecular structural and mechanical properties (biomimicry). However, the rheological behaviors of these natural-based bioinks constitute the main challenge of the cell-laden printing process (bioprinting). For this reason, bioprinting usually requires chemical modifications and/or inter-macromolecular crosslinking. In this sense, a comprehensive analysis describing these biopolymers (natural proteins and polysaccharides)-based bioinks, their modifications, and their stimuli-responsive nature is performed. This manuscript is organized into three sections: (1) tissue engineering application, (2) crosslinking, and (3) bioprinting techniques, analyzing the current challenges and strengths of biopolymers in bioprinting. In conclusion, all hydrogels try to resemble extracellular matrix properties for bioprinted structures while maintaining good printability and stability during the printing process.