Biogas represents a source of renewable energy that could provide a replacement for fossil fuels to meet the increasing demand for energy. The upgrading of biogas through the removal of CO2 to a content of 95–97% of CH4 is necessary to increase its calorific value. This review focuses on biogas upgrading technologies using wastes or residues that enable the performing of mineral carbonation. In this research, we analyzed a natural biogas or synthetic one with a content of about (40–50%) of carbon dioxide. The chemical absorption is also briefly described in this study, due to its being the first step in innovative absorption and regeneration processes using mineral carbonization. Wastes with high calcium contents, i.e., ashes, steel-making slags, and stabilized wastewater anaerobic sludge, were considered for direct carbonization, taking into account the leaching of particles from carbonated wastes/residues. Moreover, the different types of reactors used for mineral carbonation have been described. The presented technological solutions are easy to use and economical, and some of them also take into account the regeneration of reagents. However, in the context of their direct use in biogas plants, it is necessary to consider the availability of wastes and residues.