Natural bone is a composite organic-inorganic material, containing hydroxyapatite (HAP) as an inorganic phase. In this review, applications of natural alginic acid (ALGH) polymer for the fabrication of composites containing HAP are described. ALGH is used as a biocompatible structure directing, capping and dispersing agent for the synthesis of HAP. Many advanced techniques for the fabrication of ALGH-HAP composites are attributed to the ability of ALGH to promote biomineralization. Gel-forming and film-forming properties of ALGH are key factors for the development of colloidal manufacturing techniques. Electrochemical fabrication techniques are based on strong ALGH adsorption on HAP, pH-dependent charge and solubility of ALGH. Functional properties of advanced composite ALGH-HAP films and coatings, scaffolds, biocements, gels and beads are described. The composites are loaded with other functional materials, such as antimicrobial agents, drugs, proteins and enzymes. Moreover, the composites provided a platform for their loading with cells for the fabrication of composites with enhanced properties for various biomedical applications. This review summarizes manufacturing strategies, mechanisms and outlines future trends in the development of functional biocomposites.