The re-centring capability is recognized as a fundamental function of any effective isolation system, not only because it is associated to small or negligible deformation at the end of the earthquake but rather because it prevents displacement build up that may limit the capability of the structure to withstand aftershocks and future earthquakes. The current Eurocode recommends to estimate the maximum total displacement of the isolated system as the superposition of the nonseismic offset displacement resulting from permanent actions, long-term deformations and thermal movements of the structure, and of the amplified seismic displacement induced by the design earthquake. For systems endowed with low re-centring capability, the estimation shall also account for the possible accrual of displacements during the lifetime of the structure. However, the aforementioned criteria have never been evaluated for curved surface sliders, which are characterized by an inherent nonlinear behaviour.The study aims at giving more insight into the matter by conducting a parametric study based on one-directional nonlinear response time history analyses and considering a variety of seismic scenarios. The first part of the study investigates the effect of a nonseismic displacement on the earthquake-induced displacement and formulates a criterion to evaluate the capability of curved surface sliders to provide a seismic response independent of the offset displacement. The response of the isolation system to natural sequences of earthquakes, where the offset displacement is the residual displacement from the previous shake, is addressed in the second part of the paper. The provisions of the Eurocode are eventually checked against the observed data.KEYWORDS base isolation, curved surface slider, nonseismic displacement, re-centring capability, total displacement LIST OF NOTATIONS USED IN THE PAPER: d, horizontal displacement of the isolation system; D, horizontal displacement capacity of the isolator in a SDOF system; D i , horizontal displacement capacity of isolator i; d cd , design displacement at the centre of stiffness of the isolation system; d G , non-seismic displacement of the isolator in a SDOF system; d G,i , non-seismic displacement of the isolator i; d m , design seismic displacement of the isolator in a SDOF system; d m,i , design seismic displacement of the i-th isolator; d m,MS , maximum displacement induced by the main shock of a ground motion sequence; d m,o , maximum total displacement under coexisting non-seismic and seismic actions; d o , maximum design non-seismic displacement; d rm , maximum residual displacement; d r , residual displacement of the isolation system with no offset displacement; d r,o , residual displacement of the isolation system affected from the offset displacement d G ; d r,AS , residual displacement at the end of the aftershock of a ground motion sequence; d r,FS , residual displacement at the end of the foreshock of a ground motion sequence; d r,MS , residual displacement at the end of the main s...