Three-dimensional (3D) disease modeling and drug screening systems have become important in tissue engineering, drug screening, and development. The newly developed systems support cell and extracellular matrix (ECM) interactions, which are necessary for the formation of the tissue or an accurate model of a disease. Hydrogels are favorable biomaterials due to their properties: biocompatibility, high swelling capacity, tunable viscosity, mechanical properties, and their ability to biomimic the structure and function of ECM. They have been used to model various diseases such as tumors, cancer diseases, neurodegenerative diseases, cardiac diseases, and cardiovascular diseases. Additive manufacturing approaches, such as 3D printing/bioprinting, stereolithography (SLA), selective laser sintering (SLS), and fused deposition modeling (FDM), enable the design of scaffolds with high precision; thus, increasing the accuracy of the disease models. In addition, the aforementioned methodologies improve the design of the hydrogel-based scaffolds, which resemble the complicated structure and intricate microenvironment of tissues or tumors, further advancing the development of therapeutic agents and strategies. Thus, 3D hydrogel-based disease models fabricated through additive manufacturing approaches provide an enhanced 3D microenvironment that empowers personalized medicine toward targeted therapeutics, in accordance with 3D drug screening platforms.