The musculoskeletal system, consisting of bones and cartilage of various types, muscles, ligaments, and tendons, is the basis of the human body. However, many pathological conditions caused by aging, lifestyle, disease, or trauma can damage its elements and lead to severe disfunction and significant worsening in the quality of life. Due to its structure and function, articular (hyaline) cartilage is the most susceptible to damage. Articular cartilage is a non-vascular tissue with constrained self-regeneration capabilities. Additionally, treatment methods, which have proven efficacy in stopping its degradation and promoting regeneration, still do not exist. Conservative treatment and physical therapy only relieve the symptoms associated with cartilage destruction, and traditional surgical interventions to repair defects or endoprosthetics are not without serious drawbacks. Thus, articular cartilage damage remains an urgent and actual problem requiring the development of new treatment approaches. The emergence of biofabrication technologies, including three-dimensional (3D) bioprinting, at the end of the 20th century, allowed reconstructive interventions to get a second wind. Three-dimensional bioprinting creates volume constraints that mimic the structure and function of natural tissue due to the combinations of biomaterials, living cells, and signal molecules to create. In our case—hyaline cartilage. Several approaches to articular cartilage biofabrication have been developed to date, including the promising technology of 3D bioprinting. This review represents the main achievements of such research direction and describes the technological processes and the necessary biomaterials, cell cultures, and signal molecules. Special attention is given to the basic materials for 3D bioprinting—hydrogels and bioinks, as well as the biopolymers underlying the indicated products.