Three-dimensional bioprinting is a promising technology for bone tissue engineering. However, most hydrogel bioinks lack the mechanical and post-printing fidelity properties suitable for such hard tissue regeneration. To overcome these weak properties, calcium phosphates can be employed in a bioink to compensate for the lack of certain characteristics. Further, the extracellular matrix of natural bone contains this mineral, resulting in its structural robustness. Thus, calcium phosphates are necessary components of bioink for bone tissue engineering. This review paper examines different recently explored calcium phosphates, as a component of potential bioinks, for the biological, mechanical and structural properties required of 3D bioprinted scaffolds, exploring their distinctive properties that render them favorable biomaterials for bone tissue engineering. The discussion encompasses recent applications and adaptations of 3D-printed scaffolds built with calcium phosphates, delving into the scientific reasons behind the prevalence of certain types of calcium phosphates over others. Additionally, this paper elucidates their interactions with polymer hydrogels for 3D bioprinting applications. Overall, the current status of calcium phosphate/hydrogel bioinks for 3D bioprinting in bone tissue engineering has been investigated.