Water scarcity is one of the greatest societal challenges facing humanity. Reverse osmosis (RO) desalination, a widely used membrane-based technology, has proven to be effective to augment water supply in water-stressed regions of our planet. However, progress in the design and development of RO membranes has been limited. To significantly enhance the performance of RO membranes, it is essential to acquire a deep understanding of the membrane separation and transport mechanisms. In this tutorial review, we cover the pivotal historical developments in RO technology, examine the chemical and physical properties of RO membrane materials, and critically review the models and mechanisms proposed for water transport in RO membranes. Based on recent experimental and computational findings, we conduct a thorough analysis of the key transport models-the solution-diffusion and pore-flow models-to assess their validity for accurately describing water transport in RO membranes. Our analysis involves examining the experimental evidence in favor of the solution-diffusion mechanism. Specifically, we explain whether the water content gradient within the membrane, cited as evidence for the key assumption in the solution-diffusion model, can drive a diffusive transport through RO membranes. Additionally, we review the recent molecular dynamics simulations which support the pore-flow mechanism for describing water transport in RO membranes. We conclude by providing future research directions aimed at addressing key knowledge gaps in water transport phenomena in RO membranes, with the goal of advancing the development of next-generation RO membranes.