In connection with questions concerning the resonant behaviour of certain types of offshore structures, the response of non-linearly damped systems to Gaussian excitation has been studied. Both a single and a multi-degree of freedom system excited predominantly at one natural frequency have been considered. Explicit descriptions of the response, partly in the time-domain and partly in terms of non-Gaussian probability distributions, have been derived for the limit in case the damping is small. The solutions have been used to indicate the effect of non-linear damping on the distribution of response maxima, on the expected fatigue damage and on the expected extreme response. Comparisons have been made with results obtained from equivalent linearisation methods and with results obtained from numerical time-domain simulations of the non-linearly damped response of a marine riser to random waves.