Fluid mechanics lies at the heart of many of the physical processes associated with the nascent deep-sea mining industry. The evolution and fate of sediment plumes that would be produced by seabed mining activities, which are central to the assessment of the environmental impact, are entirely determined by transport processes. These processes, which include advection, turbulent mixing, buoyancy, differential particle settling, and flocculation, operate at a multitude of spatiotemporal scales. A combination of historical and recent efforts that combine theory, numerical modeling, laboratory experiments, and field trials has yielded significant progress, including assessing the role of environmental and operational parameters in setting the extent of sediment plumes, but more fundamental and applied fluid mechanics research is needed before models can accurately predict commercial-scale scenarios. Furthermore, fluid mechanics underpins the design and operation of proposed mining technologies, for which there are currently no established best practices. Expected final online publication date for the Annual Review of Fluid Mechanics, Volume 55 is January 2023. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.