An asymptotic analysis of two-dimensional free-surface cusps associated with flows at low Reynolds numbers is presented on the basis of a model which, in agreement with direct experimental observations, considers this phenomenon as a particular case of an interface formation–disappearance process. The model was derived from first principles and earlier applied to another similar process: the moving contact-line problem. As is shown, the capillary force acting on a cusp from the free surface, which in the classical approach can be balanced by viscous stresses only if the associated rate of dissipation of energy is infinite, in the present theory is always balanced by the force from the surface-tension-relaxation ‘tail’, which stretches from the cusp towards the interior of the fluid. The flow field near the cusp is shown to be regular, and the surface-tension gradient in the vicinity of the cusp, caused and maintained by the external flow, induces and is balanced by the shear stress. Existing approaches to the free-surface cusp description and some relevant experimental aspects of the problem are discussed.