Placing mechanical devices into fast-moving tidal streams to generate clean and predictable electricity is a developing technology. This review covers the fundamental fluid mechanics of this application, which is important for understanding how such devices work and how they interact with the tidal stream resource. We focus on how tidal stream turbines and energy generation are modeled analytically, numerically, and experimentally. Owing to the nature of the problem, our review is split into different scales—from turbine to array and regional—and we examine each in turn. Expected final online publication date for the Annual Review of Fluid Mechanics, Volume 53 is January 6, 2021. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.