Every year flood events lead to thousands of casualties and significant economic damage. Mapping the areas at risk of flooding is critical to reducing these losses, yet until the last few years such information was available for only a handful of well-studied locations. This review surveys recent progress to address this fundamental issue through a novel combination of appropriate physics, efficient numerical algorithms, high-performance computing, new sources of big data, and model automation frameworks. The review describes the fluid mechanics of inundation and the models used to predict it, before going on to consider the developments that have led in the last five years to the creation of the first true fluid mechanics models of flooding over the entire terrestrial land surface.

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