Surface water storage and fluxes in rivers, lakes, reservoirs and wetlands are currently poorly observed at the global scale, even though they represent major components of the water cycle and deeply impact human societies. In situ networks are heterogeneously distributed in space, and many river basins and most lakes—especially in the developing world and in sparsely populated regions—remain unmonitored. Satellite remote sensing has provided useful complementary observations, but no past or current satellite mission has yet been specifically designed to observe, at the global scale, surface water storage change and fluxes. This is the purpose of the planned Surface Water and Ocean Topography (SWOT) satellite mission. SWOT is a collaboration between the (US) National Aeronautics and Space Administration, Centre National d'Études Spatiales (the French Spatial Agency), the Canadian Space Agency and the United Kingdom Space Agency, with launch planned in late 2020. SWOT is both a continental hydrology and oceanography mission. However, only the hydrology capabilities of SWOT are discussed here. After a description of the SWOT mission requirements and measurement capabilities, we review the SWOT-related studies concerning land hydrology published to date. Beginning in 2007, studies demonstrated the benefits of SWOT data for river hydrology, both through discharge estimation directly from SWOT measurements and through assimilation of SWOT data into hydrodynamic and hydrology models. A smaller number of studies have also addressed methods for computation of lake and reservoir storage change or have quantified improvements expected from SWOT compared with current knowledge of lake water storage variability. We also briefly review other land hydrology capabilities of SWOT, including those related to transboundary river basins, human water withdrawals and wetland environments. Finally, we discuss additional studies needed before and after the launch of the mission, along with perspectives on a potential successor to SWOT.