Data are presented on the vibration of aeroplane structures excited by jet-noise or other random pressures. A review is made of those aspects of vibration and transmitted noise which may be alleviated by increasing the structural damping. The damping of conventional untreated structures is considered, measured values of the damping of model and full-scale structures are discussed in the light of previous studies of acoustic and rivet damping. The common practice of judging a damping treatment by the loss factor increment it produces is shown to be inadequate for contemporary treatments and thin-plate structures. Alternative criteria are derived which describe more adequately the effectiveness of the treatment in attenuating certain random and harmonic vibration phenomena. These were used to compare two commercial unconstrained treatments and to estimate the reduction of random stresses in an aeroplane structure when the treatments are added. The response of a two-dimensional flat sandwich plate with a damped core was analysed. The dependence of the modal loss factor, stiffness and criteria values on the wavelength and core properties was studied.