Three-dimensional (3D) printing, also known as additive manufacturing, has revolutionized the production of physical 3D objects by transforming computer-aided design models into layered structures, eliminating the need for traditional molding or machining techniques. In recent years, hydrogels have emerged as an ideal 3D printing feedstock material for the fabrication of hydrated constructs that replicate the extracellular matrix found in endogenous tissues. Hydrogels have seen significant advancements since their first use as contact lenses in the biomedical field. These advancements have led to the development of complex 3D-printed structures that include a wide variety of organic and inorganic materials, cells, and bioactive substances. The most commonly used 3D printing techniques to fabricate hydrogel scaffolds are material extrusion, material jetting, and vat photopolymerization, but novel methods that can enhance the resolution and structural complexity of printed constructs have also emerged. The biomedical applications of hydrogels can be broadly classified into four categories—tissue engineering and regenerative medicine, 3D cell culture and disease modeling, drug screening and toxicity testing, and novel devices and drug delivery systems. Despite the recent advancements in their biomedical applications, a number of challenges still need to be addressed to maximize the use of hydrogels for 3D printing. These challenges include improving resolution and structural complexity, optimizing cell viability and function, improving cost efficiency and accessibility, and addressing ethical and regulatory concerns for clinical translation.