Non-expert summaryThis is the first paper to present a coherent treatment of the formation of a hydraulic jump by a smooth, coherent liquid film impinging normally on to a smooth horizontal plate. The liquid spreads out radially in a thin layer bounded by a circular hydraulic jump, outside which the depth is much greater. The motion in the layer is analyzed in terms of boundary-layer theory, both for laminar and for turbulent flow, and relations are obtained for the radius of the hydraulic jump. These relations are compared with experimental results obtained for water jets in air.. The analogous problems of two-dimensional flow are also considered.