The mechanism for the generation of vorticity at a viscous free surface is described. This is a free-surface analogue of Lighthill's strategy for determining the vorticity flux at solid boundaries. In this method the zero-shear-stress and pressure boundary conditions are transformed into a boundary integral formulation suitable for the velocity–vorticity description of the flow. A vortex sheet along the free surface is determined by the pressure boundary condition, while the condition of zero shear stress determines the vorticity at the surface. In general, vorticity is generated at free surfaces whenever there is flow past regions of surface curvature. It is shown that vorticity is conserved in free-surface viscous flows. Vorticity which flows out of the fluid across the free surface is gained by the vortex sheet; the integral of vorticity over the entire fluid region plus the integral of ‘surface vorticity’ over the free surface remains constant. The implications of the present strategy as an algorithm for numerical calculations are discussed.