Hydroxyapatite (HAp) has been considered for decades an ideal biomaterial for bone repair due to its compositional and crystallographic similarity to bioapatites in hard tissues. However, fabrication of porous HAp acting as a template (scaffold) for supporting bone regeneration and growth has been a challenge to biomaterials scientists. The introduction of additive manufacturing technologies, which provide the advantages of a relatively fast, precise, controllable, and potentially scalable fabrication process, has opened new horizons in the field of bioceramic scaffolds. This review focuses on three-dimensional-printed HAp scaffolds and related composite systems, where the calcium phosphate phase is combined with other ceramics or polymers improving the mechanical properties and/or imparting special extra-functionalities. The main applications of three-dimensional-printed HAp scaffolds in bone tissue engineering are presented and discussed; furthermore, this review also emphasizes the most recent achievements toward the development and testing of multifunctional HAp-based systems combining multiple properties for advanced therapy (e.g., bone regeneration, antibacterial effect, angiogenesis, and cancer treatment).