This review paper examines the applicability of biogas and biomethane as potential maritime fuels and examines issues of these fuels from a supply chain perspective (from production to end use). The objectives are to identify: (1) the latest research, development, and innovation activities; (2) issues and key barriers related to the technology readiness to bring biogas/biomethane to market; and (3) commercialisation issues, including cost parity with natural gas (the main competitor). A survey of the literature was carried out based on research articles and grey literature. The PESTEL and SWOT analyses identified opportunities for these fuels due to the relevant regulations (e.g., Fit for 55; the recent inclusion of the Mediterranean Sea as a SECA and PM control area; MPEC 79), market-based measures, and environmental, social, and governance strategies. The potential of biomass feedstock is estimated to have a substantial value that can satisfy the energy needs of the maritime industry. However, production costs of biomethane are high; estimated to be 2–4 times higher compared to natural gas. The market is moving in the direction of alternative drop-in fuels, including liquefied and compressed biomethane (LBM and CBM) and biogas. In terms of potential market penetration, LBM can be used as a marine drop-in fuel for the existing fleet that already combust LNG and LPG due to similar handling. Currently, these vessels are LNG and LPG tankers. However, in newly built vessels, LBM can be also supplied to container ships, vehicle carriers, and bulk carriers (about 20% of newly built vessels). Provided that compressed natural gas infrastructure exists, CBM can be exploited in vessels with low energy needs and low space requirements and shore-side electrification, because investments in retrofits are lower compared to constructing new infrastructure.