Hydrogen-based technologies are among the most promising solutions to fulfill the zero-emission scenario and ensure the energy independence of many countries. Hydrogen is considered a green energy carrier, which can be utilized in the energy, transport, and chemical sectors. However, efficient and safe large-scale hydrogen storage is still challenging. The most frequently used hydrogen storage solutions in industry, i.e., compression and liquefaction, are highly energy-consuming. Underground hydrogen storage is considered the most economical and safe option for large-scale utilization at various time scales. Among underground geological formations, salt caverns are the most promising for hydrogen storage, due to their suitable physicochemical and mechanical properties that ensure safe and efficient storage even at high pressures. In this paper, recent advances in underground storage with a particular emphasis on salt cavern utilization in Europe are presented. The initial experience in hydrogen storage in underground reservoirs was discussed, and the potential for worldwide commercialization of this technology was analyzed. In Poland, salt deposits from the north-west and central regions (e.g., Rogóźno, Damasławek, Łeba) are considered possible formations for hydrogen storage. The Gubin area is also promising, where 25 salt caverns with a total capacity of 1600 million Nm3 can be constructed.