In recent years, a growing demand for biomaterials has been observed, particularly for applications in bone regenerative medicine. Bone tissue engineering (BTE) aims to develop innovative materials and strategies for repairing and regenerating bone defects and injuries. Polysaccharides, due to their biocompatibility, biodegradability as well as bioactivity, have emerged as promising candidates for scaffolds or composite systems in BTE. Polymers combined with bioactive ceramics can support osteointegration. Calcium phosphate (CaP) ceramics can be a broad choice as an inorganic phase that stimulates the formation of new apatite layers. This review provides a comprehensive analysis of composite systems based on selected polysaccharides used in bone tissue engineering, highlighting their synthesis, properties and applications. Moreover, the applicability of the produced biocomposites has been analyzed, as well as new trends in modifying biomaterials and endowing them with new functionalizations. The effects of these composites on the mechanical properties, biocompatibility and osteoconductivity were critically analyzed. This article summarizes the latest manufacturing methods as well as new developments in polysaccharide-based biomaterials for bone and cartilage regeneration applications.