The desulphurisation of biogas for hydrogen sulphide (H2S) removal constitutes a significant challenge in the area of biogas research. This is because the retention of H2S in biogas presents negative consequences on human health and equipment durability. The negative impacts are reflective of the potentially fatal and corrosive consequences reported when biogas containing H2S is inhaled and employed as a boiler biofuel, respectively. Recognising the importance of producing H2S-free biogas, this paper explores the current state of research in the area of desulphurisation of biogas. In the present paper, physical–chemical, biological, in-situ, and post-biogas desulphurisation strategies were extensively reviewed as the basis for providing a qualitative comparison of the strategies. Additionally, a review of the costing data combined with an analysis of the inherent data uncertainties due underlying estimation assumptions have also been undertaken to provide a basis for quantitative comparison of the desulphurisation strategies. It is anticipated that the combination of the qualitative and quantitative comparison approaches employed in assessing the desulphurisation strategies reviewed in the present paper will aid in future decisions involving the selection of the preferred biogas desulphurisation strategy to satisfy specific economic and performance-related targets.