In tissue engineering, the fate of a particular organ/tissue regeneration and repair mainly depends on three pillars - 3D architecture, cells used, and stimulus provided. 3D cell supportive structure development is one of the crucial pillars necessary for defining organ/tissue geometry and shape. In recent years, the advancements in 3D bio-printing (additive manufacturing) made it possible to develop very precise 3D architectures with the help of industrial software like Computer-Aided Design (CAD). The main requirement for the 3D printing process is the bio-ink, which can act as a source for cell support, proliferation, drug (growth factors, stimulators) delivery, and organ/tissue shape. The selection of the bio-ink depends upon the type of 3D tissue of interest. Printing tissues like bone and cartilage is always challenging because it is difficult to find printable biomaterial that can act as bio-ink and mimic the strength of the natural bone and cartilage tissues. This review describes different biomaterials used to develop bio-inks with different processing variables and cell-seeding densities for bone and cartilage 3D printing applications. The review also discusses the advantages, limitations, and cell bio-ink compatibility in each biomaterial section. The emphasis is given to bio-inks reported for 3D printing cartilage and bone and their applications in orthopedics and orthodontists. The critical/important performance and the architectural morphology requirements of desired bone and cartilage bio-inks were compiled in summary.