Ice sheets are expected to shrink in size as the world warms, which in turn will raise sea level. The West Antarctic ice sheet is of particular concern, because it was probably much smaller at times during the past million years when temperatures were comparable to levels that might be reached or exceeded within the next few centuries. Much of the grounded ice in West Antarctica lies on a bed that deepens inland and extends well below sea level. Oceanic and atmospheric warming threaten to reduce or eliminate the floating ice shelves that buttress the ice sheet at present. Loss of the ice shelves would accelerate the flow of non-floating ice near the coast. Because of the slope of the sea bed, the consequent thinning could ultimately float much of the ice sheet's interior. In this scenario, global sea level would rise by more than three metres, at an unknown rate. Simplified analyses suggest that much of the ice sheet will survive beyond this century. We do not know how likely or inevitable eventual collapse of the West Antarctic ice sheet is at this stage, but the possibility cannot be discarded. For confident projections of the fate of the ice sheet and the rate of any collapse, further work including the development of well-validated physical models will be required.