This review focuses on Leonardo da Vinci's work and thought related to fluid mechanics as it is presented in a lifetime of notebooks, letters, and artwork. It shows how Leonardo's remaining works offer a complicated picture of unfinished, scattered, and frequently revisited hypotheses and conclusions. It argues that experimentation formed an important mechanism for Leonardo's thought about natural fluid flows, which was an innovation to the scientific thinking of his day, but which did not always lead him to the conclusions of modern fluid mechanics. It highlights the multiple and ambiguous meanings of turbulence in his works. It examines his thinking suggestive of modern concepts such as the no-slip condition, hydraulic jump, cardiovascular vortices, conservation of volume, and the distinctive path of ascending bubbles we now term Leonardo's paradox, among others. It demonstrates how Leonardo thought through analogies, building-block flow patterns, and synthesis, leading both to successes—especially in the management of water—and to failures, perhaps most obviously in his pursuit of human flight.