We collect data on the movement and productivity of elite scientists. Their mobility is remarkable: nearly half of the world's most-cited physicists work outside their country of birth. We show they migrate systematically towards nations with large R&D spending. Our study cannot adjudicate on whether migration improves scientists' productivity, but we find that movers and stayers have identical h-index citations scores. Immigrants in the UK and US now win Nobel Prizes proportionately less often than earlier. US residents' h-indexes are relatively high. We describe a framework where a key role is played by low mobility costs in the modern world.