At first glance, the migration pressures on the EU and US appear similar, but recent history is not a reliable guide to future trends. This column uses demographic trends to predict that the US will experience a gradual decline in its newly arrived immigrant population, while the EU, ringed by nearby high-population-growth states, will see large increases in the stock of first-generation immigrants. As a result, US emphasis on strengthening borders and returning undocumented migrants may be misplaced.