New experiments were performed with three-dimensional thermomechanical models of the Greenland and Antarctic ice sheets to simulate their behaviour during the glacial cycles, to reconstruct their thickness and extent at the Last Glacial Maximum, and to estimate their glacio-eustatic contribution to the global sea level stand. The calculations used improved ice-dynamic and isostatic treatments, updated datasets on higher grid resolutions, and refined climatic treatments based on newly calibrated transfer functions between ice core records and climatic perturbations. Results are discussed from a reference run with standard parameters that is compared with available glacial-geological observations, and from a series of sensitivity experiments focusing on isostatic adjustment, thermomechanical coupling, climatic forcing, mass-balance changes, and basal melting rates and viscosity changes of Antarctic ice shelves. For the Antarctic ice sheet, we find that volume changes are closely linked with grounding line changes of the West Antarctic ice sheet. At the LGM, the grounding line extended close to the continental shelf break almost everywhere. Ice over central East Antarctica was generally thinner than today and varied mainly in accordance with accumulation fluctuations. For the Greenland ice sheet, melting is important only during interglacial periods and the most sensitive period concerns the size of the ice sheet during the Eemian. At the LGM, the Greenland ice sheet extended beyond the present coastline to cover at least the inner continental shelf and thinned by up to several hundred meters in central areas. For a plausible range of parameters, the experiments indicate that at the time of maximum sea-level depression (21 kyr BP), the Antarctic ice sheet contributed 14-18 m to the sea-level lowering, and the Greenland ice sheet 2-3 m, significantly less than the older CLIMAP reconstructions. Whereas both ice sheets were at 21 kyr BP close to their maximum extent, the experiments also indicate that their maximum volumes were reached only by 16.5 kyr BP (Greenland) and 10 kyr BP (Antarctica), equal to an additional sea-level lowering of respectively 0.4 m and 3.7 m. Holocene retreat was essentially complete by 5 kyr BP in Greenland, but is found to still continue today in West Antarctica before reversing to growth during the next millenium. The models were found to reproduce gross features of the ice sheets history since the LGM in reasonably good agreement with available glacial-geological data, although observational control on ice thickness changes remains very limited.