A theory for high-temperature superconductivity is developed based on a hydrodynamic approximation to the many-body problem. Electrons confined to two-dimensional sheets exchange coupled two-dimensional plasmon and phonon modes. This leads to a strong electron-electron attraction. Bardeen-Cooper-Schrieffer theory is used to derive an expression for the transition temperature. The superconductivity comes about because of the existence of a low-frequency mode. Experimental consequences of the theory are discussed.