Recovery of the energy contained in biogas will be essential in coming years to reduce greenhouse gas emissions and our current dependence on fossil fuels. The elimination of H₂S is a priority to avoid equipment corrosion, poisoning of catalytic systems and SO₂ emissions in combustion engines. This review describes the advances made in this technology using fixed biomass bioreactors (FBB) and suspended growth bioreactors (SGB) since the first studies in this field in 2008. Anoxic desulfurization has been studied mainly in biotrickling filters (BTF). Elimination capacities (EC) up to 287 gS m^-3 h^-1 have been achieved, with a removal efficiency (RE) of 99%. Both nitrate and nitrite have been successfully used as electron acceptor. SGBs can solve some operational problems present in FBBs, such as clogging or nutrient distribution issues. However, they present greater difficulties in gas-liquid mass transfer, although ECs of up to 194 gS m^-3 h^-1 have been reported in both gas-lift and stirred tank reactors. One of the major disadvantages of using anoxic biodesulfurization compared to aerobic biodesulfurization is the need to provide reagents (nitrates and/or nitrites), with the consequent increase in operating costs. A solution proposed in this respect is the use of nitrified effluents, some ammonium-rich effluents nitrified include landfill leachate and digested effluent from the anaerobic digester have been tested successfully. Among the microbial diversity found in the bioreactors, the genera Thiobacillus, Sulfurimonas and Sedimenticola play a key role in anoxic removal of H₂S. Finally, a summary of future trends in technology is provided.