Marine carbon storage plays a crucial role in reducing global greenhouse gas emissions. To ensure the efficient and safe storage of CO₂, it is imperative to monitor the potential migration of CO₂ before, during, and after injection. Current methods for monitoring marine carbon storage encompass built-in sensor monitoring focusing on the seabed wellbore, geophysical monitoring targeting reservoirs and caprocks, and marine environmental monitoring focusing on the seafloor and water column. These three methods can be used to obtain temperature/pressure/acoustic data near the injection/monitoring wellbore, seismic/electromagnetic/gravity data of deep reservoirs and caprocks, and acoustic/chemical/oceanographic data of near-bottom sedimentary layers and seawater, respectively. Analyzing these datasets is expected to reveal the migration characteristics of CO₂ injected into the formation. However, the integrated use of relevant monitoring methods and technologies and the design of high-quality monitoring strategies currently pose significant challenges for both academic and engineering communities. To enable scientific and systematic monitoring of the safety of marine carbon storage, offering essential guidance for offshore storage operations, and concurrently enhancing monitoring efficiency while reducing monitoring costs, we have compiled the fundamental principles, application status, and challenges encountered by different monitoring methods and technologies. We also anticipate future development of monitoring technologies for marine carbon storage.