The paper presents the results of a study in which the non-stationary dynamic response of typical lift installations is investigated. A general approach to describe the dynamic behaviour of a vertical transport installation is presented. Subsequently, vibration models of a building elevator and a mine hoist installation are discussed. Perturbation and numerical techniques are discussed and applied to predict the non-stationary response of hoist ropes. It is shown that the method of multiple scales with non-linear scale as well as the method of characteristics can be employed to analyse a passage through resonance in a simple lift installation. Furthermore, the effectiveness of direct numerical integration of equations of motion is demonstrated in the case of a mine hoist installation.