Renewable energy production is limited by the fluctuations limiting their application. Underground hydrogen storage (UHS) is one possible alternative to reduce the gap between supply and demand by storing the energy converted to hydrogen as a carrier and store it during surplus to produce it during high demand periods. The hydrogen is stored in the subsurface in geological formations containing the gas and is injected/produced via wells. There is a lack of experience associated with this technology and only a small number of projects worldwide. There are several mechanisms that can compromise the integrity of the well and generate leakage of the stored gas. This paper aims to introduce the challenges associated with well integrity of UHS. Mechanisms that can compromise well integrity and generate leaks include microbial corrosion, hydrogen blistering hydrogen-induced cracking and hydrogen embrittlement, cement degradation, elastomer failure, and caprock sealing failure. Propose well completion criteria, recommendation, and materials selection for newly constructed wells or existing wells. A comparison with more developed storage technologies aims to provide a better understanding of the limitations of hydrogen storage by comparing it to carbon dioxide (carbon capture and storage) and methane (underground gas storage). Finally, evaluation and monitoring techniques are required to see the influence of hydrogen on well integrity. Future research and development will reduce the uncertainties and limitations associated with UHS increasing its feasibility and implementation.