I review the origins and development of the idea of Dyson spheres, their purpose, their engineering, and their detectability. I explicate the ways in which the popular imagining of them as monolithic objects would make them dynamically unstable under gravity and radiation pressure, and mechanically unstable to buckling. I develop a model for the radiative coupling between a star and large amounts of material orbiting it, and connect the observational features of a star plus Dyson sphere system to the gross radiative properties of the sphere itself. I discuss the still-unexplored problem of the effects of radiative feedback on the central star's structure and luminosity. Finally, I discuss the optimal sizes of Dyson spheres under various assumptions about their purpose as sources of low-entropy emission, dissipative work, or computation.