Marine-terminating glaciers control most of Greenland's ice discharge into the ocean, but little is known about the geometry of their frontal regions. Here we use side-looking, multibeam echo sounding observations to reveal that their frontal ice cliffs are grounded deeper below sea level than previously measured and their ice faces are neither vertical nor smooth but often undercut by the ocean and rough. Deep glacier grounding enables contact with subsurface, warm, salty Atlantic waters (AW) which melts ice at rates of meters per day. We detect cavities undercutting the base of the calving faces at the sites of subglacial water (SGW) discharge predicted by a hydrological model. The observed pattern of undercutting is consistent with numerical simulations of ice melt in which buoyant plumes of SGW transport warm AW to the ice faces. Glacier undercutting likely enhances iceberg calving, impacting ice front stability and, in turn, the glacier mass balance.