This paper develops a systematic theory for the flow observed in the so-called "overflowing cylinder" experiment. The basic phenomenon to be explained is the order of magnitude increase in the surface velocity of a slowly overflowing beaker of water that is caused by the addition of a small amount of soluble surfactant. We perform analyses of (i) an inviscid bulk flow in which diffusion is negligible, (ii) a hydrodynamic boundary layer in which viscous effects become important, and (iii) a diffusive boundary layer where diffusion is significant, and by matching these together arrive at a coupled problem for the liquid velocity and surfactant concentration. Our model predicts a relation between surface velocity and surface concentration which is in good agreement with experiment. However, a degeneracy in the boundary conditions leaves one free parameter which must be taken from experimental data. We suggest an investigation that may resolve this indeterminacy.