This paper examines the elastic and inelastic seismic behaviour of single layer steel cylindrical lattice shells. The main dynamic characteristics for this form of structure are firstly examined through a parametric assessment, which also leads to proposed expressions for estimating the fundamental period and mode of vibration. The seismic response of five typical shell configurations, representing a wide range of rise to span ratios, is then assessed within the linear elastic range under selected earthquake excitations. Particular focus is given to the relative influence of the horizontal and vertical seismic components on the internal actions. In order to provide a means for evaluating the underlying inelastic behaviour, a simple pushover approach, which is suitable for this structural form, is suggested using the forces obtained from the fundamental mode shape. The peak angle change is proposed as a damage parameter within the nonlinear analysis for characterising the inelastic global and local demands in shells of different geometries. Incremental dynamic analysis is subsequently carried out in order to evaluate the detailed nonlinear time history response. The results provide detailed insights into the influence of the horizontal and vertical excitations on the nonlinear seismic response, and illustrate the suitability of the peak angle change as an inelastic deformation measure for shells of different geometric configurations. The main findings from the linear and nonlinear assessments are highlighted within the discussions, with a view to providing guidance for performance based assessment procedures as well as simplified design approaches.