We propose a rigorous solution to a long-standing problem of the impact of eddy currents on the dispersion relation of surface spin waves propagating in thin conducting magnetic films. Our results confirm the prediction of the Almeida-Mill's exchange-free theory that the inclusion of the eddy-current contribution results in a deviation of the dispersion curve for the fundamental mode from the Damon-Eshbach law and a substantial linewidth broadening in a large wave vector range. We show that the decrease in the spin-wave frequency is due to an increase in the in-plane component of the dynamic magnetic field within the conducting film. The decrease in the frequency is accompanied by a drastic change in the asymmetry of the modal profiles for the waves. This effect is not observable in magneto-insulating films and therefore it is unambiguously attributed to eddy currents that appear in conducting films only. We also show that the wave vector range in which eddy currents affect the dispersion curve is strongly correlated with the value of the film conductivity. This result holds for conducting films with the thickness 10-100 nm, which are considered promising for future magnonic and spintronic applications. © 2013 IOP Publishing Ltd.