Additive manufacturing (AM), frequently cited as three-dimensional (3D) printing, is a relatively new manufacturing technique for biofabrication, also called 3D manufacture with biomaterials and cells. Recent advances in this field will facilitate further improvement of personalized healthcare solutions. In this regard, tailoring several healthcare products such as implants, prosthetics, and in vitro models, would have been extraordinarily arduous beyond these technologies. Three-dimensional-printed structures with a multiscale porosity are very interesting manufacturing processes in order to boost the capability of composite scaffolds to generate bone tissue. The use of biomimetic hydroxyapatite as the main active ingredient for bioinks is a helpful approach to obtain these advanced materials. Thus, 3D-printed biomimetic composite designs may produce supplementary biological and physical benefits. Three-dimensional bioprinting may turn to be a bright solution for regeneration of bone tissue as it enables a proper spatio-temporal organization of cells in scaffolds. Different types of bioprinting technologies and essential parameters which rule the applicability of bioinks are discussed in this review. Special focus is made on hydroxyapatite as an active ingredient for bioinks design. The goal of such bioinks is to reduce the constraints of commonly applied treatments by enhancing osteoinduction and osteoconduction, which seems to be exceptionally promising for bone regeneration.