The rate of damping of surface gravity–capillary waves is investigated, in a system which consists of a thin layer of a Newtonian viscous fluid of thicknessdfloating on a Newtonian fluid of infinite depth. The surface and interfacial tensions, elasticities and viscosities are taken into account. In particular, an approximate dispersion relation is derived for the case wherekdand(&ohgr;/&ngr;+)1/2dare both small, wherekis the wavenumber,&ohgr;is the angular frequency and&ngr;+is the kinematic viscosity of the upper fluid. Ifd→0while&ngr;+dremains finite, published dispersion relations for viscoelastic surface films of extremely small (e.g., monomolecular) thickness are reproduced, if we add the surface and interfacial tensions, elasticities and viscosities together, and then add an additional4&rgr;+&ngr;+dto the surface viscosity, where&rgr;+is the density of the upper fluid. A simple approximation is derived for the damping rate and associated frequency shift when their magnitudes are both small. An example is given of what may happen with a slick of heavy fuel oil on water: a slick 10&mgr;mthick produces a damping rate only slightly different from that of a film of essentially zero thickness, but the effect of the finite thickness becomes very noticeable if it is increased to 0.1–1 mm. ©1997 American Institute of Physics.