This paper presents the bed topography of Jakobshavn Isbræ, Greenland, and Byrd Glacier, Antarctica, derived from sounding these glaciers with high-sensitivity radars. To understand the processes causing the speed-up and retreat of outlet glaciers, and to enable the development of next-generation ice-sheet models, we need information on bed topography and basal conditions. To this end, we performed measurements with the progressively improved Multichannel Coherent Radar Depth Sounder/Imager (MCoRDS/I). We processed the data from each antenna-array element using synthetic aperture radar algorithms to improve radar sensitivity and reduce along-track surface clutter. We then applied array and image-processing algorithms to extract the weak bed echoes buried in off-vertical scatter (cross-track surface clutter). At Jakobshavn Isbræ, we observed 2.7 km thick ice ~30 km upstream of the calving front and ~850 m thick ice at the calving front. We also observed echoes from multiple interfaces near the bed. We applied the MUSIC algorithm to the data to derive the direction of arrival of the signals. This analysis revealed that clutter is dominated by the ice surface at Jakobshavn Isbræ. At Byrd Glacier, we found ~3.62 km thick ice, as well as a subglacial trench ~3.05 km below sea level. We used ice thickness information derived from radar data in conjunction with surface elevation data to generate bed maps for these two critical glaciers. The performance of current radars must be improved further by ~15 dB to fully sound the deepest part of Byrd Glacier. Unmanned aerial systems equipped with radars that can be flown over lines spaced as close as 5 m apart in the cross-track direction to synthesize a two-dimensional aperture would be ideal for collecting fine-resolution data over glaciers like Jakobshavn near their grounding lines.