Surface meltwater drains across ice sheets, forming melt ponds that can trigger ice-shelf collapse(1,2), acceleration of grounded ice flow and increased sea-level rise(3-5). Numerical models of the Antarctic Ice Sheet that incorporate meltwater's impact on ice shelves, but ignore the movement of water across the ice surface, predict a metre of global sea-level rise this century(5) in response to atmospheric warming(6). To understand the impact of water moving across the ice surface a broad quantification of surface meltwater and its drainage is needed. Yet, despite extensive research in Greenland(7-10) and observations of individual drainage systems in Antarctica(10-17), we have little understanding of Antarctic-wide surface hydrology or how it will evolve. Here we show widespread drainage of meltwater across the surface of the ice sheet through surface streams and ponds (hereafter 'surface drainage') as far south as 85 degrees S and as high as 1,300 metres above sea level. Our findings are based on satellite imagery from 1973 onwards and aerial photography from 1947 onwards. Surface drainage has persisted for decades, transporting water up to 120 kilometres from grounded ice onto and across ice shelves, feeding vast melt ponds up to 80 kilometres long. Large-scale surface drainage could deliver water to areas of ice shelves vulnerable to collapse, as melt rates increase this century. While Antarctic surface melt ponds are relatively well documented on some ice shelves, we have discovered that ponds often form part of widespread, large-scale surface drainage systems. In a warming climate, enhanced surface drainage could accelerate future ice-mass loss from Antarctic, potentially via positive feedbacks between the extent of exposed rock, melting and thinning of the ice sheet.