This paper presents indicative results from the numerical investigation of two special issues of the seismic behaviour of base-isolated buildings, using custom-made software that utilizes modern objectoriented design approaches. The first issue concerns the modelling of the nonlinear behaviour of seismic isolation systems, focusing on the lead rubber bearings (LRBs), which are among the most commonly used seismic isolation systems. In particular, the inaccuracies between the actual behaviour of the LRBs, which can be more precisely represented by the Bouc-Wen model, and the usage of a bilinear inelastic model, which is often used in practice, are assessed through numerical simulations and parametric analyses. The second issue concerns potential pounding of base-isolated buildings with adjacent structures, when the available clearance around a seismically isolated building is limited, during very strong earthquakes. The consequences of potential pounding and the influence of certain parameters on the overall seismic response of base-isolated buildings are also assessed through numerical simulations and parametric analyses using custom-made software.