During the past decade, there has been a continued increase in the demand for bone defect repair and replacement resulting from long-term illnesses or traumatic incidents. To address these challenges, tissue engineering research has focused on biomedical applications. This field concentrated on the development of suitable materials to enhance biological functionality and bone integration. Toward this aim, it is necessary to develop a proper material that provides good osseointegration and mechanical behavior by combining biopolymers with ceramics, which increase their mechanical stability and mineralization process. Hydroxyapatite (HAp) is synthesized from natural resources owing to its unique properties; for example, it can mimic the composition of bones and teeth of humans and animals. Biopolymers, including chitosan and alginate, combined with HAp, offer good chemical stability and strength required for tissue engineering. Composite biomaterials containing hydroxyapatite could be a potential substitute for artificial synthetic bone grafts. Utilizing various polymers and fabrication methodologies would efficiently customize physicochemical properties and suitable mechanical properties in synergy with biodegradation, thus enhancing their potential in bone regeneration. This review summarizes the commonly used polymers in tissue engineering, emphasizing their advantages and limitations. This paper also highlights recent advances in the production and investigation of HAp-based polymer composites used in biomedical applications.