Improvements in Tissue Engineering and Regenerative Medicine (TERM)–type technologies have allowed the development of specific materials that, together with a better understanding of bone tissue structure, have provided new pathways to obtain biomaterials for bone tissue regeneration. In this manuscript, bioabsorbable materials are presented as emerging materials in tissue engineering therapies related to bone lesions because of their ability to degrade in physiological environments while the regeneration process is completed. This comprehensive review aims to explore the studies, published since its inception (2010s) to the present, on bioabsorbable composite materials based on PLA and PCL polymeric matrix reinforced with Mg, which is also bioabsorbable and has recognized osteoinductive capacity. The research collected in the literature reveals studies based on different manufacturing and dispersion processes of the reinforcement as well as the physicochemical analysis and corresponding biological evaluation to know the osteoinductive capacity of the proposed PLA/Mg and PCL/Mg composites. In short, this review shows the potential of these composite materials and serves as a guide for those interested in bioabsorbable materials applied in bone tissue engineering.