The large ice sheets in Greenland and Antarctica are losing mass due to global warming. In particular, the acceleration of ice streams and thus the increased discharge into the ocean contributes significantly to global sea-level rise. The floating extensions of the ice streams counteract this, but intense basal melting can destabilise the ice shelves. In this thesis, a contribution is made to determine the melt rates of two ice shelves, which are crucial for the future mass losses of the respective ice sheets. In the north, the focus is on the Northeast Greenland Ice Stream (NEGIS) that feeds the Nioghalvfjerdsbrae (79°N Glacier). My analysis of phase-sensitive radar measurements indicates high melt rates near the onset of the ice stream and thus the presence of subglacial melt water, which is associated with the formation of the ice flow. An extensive study in my thesis reveals that the 79°N Glacier has been thinned out considerably in recent years due to extreme melt rates and that large channels have been formed. Melt rates of the Filchner Ice Shelf, Antarctica, which I also determined using phase-sensitive radar measurements, are comparatively low. I was able to attribute significant deviations from remote sensing-derived melt rates to inaccuracies in the used ice flow velocity field. Furthermore, I show that the use of newer velocity fields improves the determination of the melt rates from remote sensing. My analysis of melt rate time series in the vicinity of a channel indicates higher melt rates in the summer as well as several melt events spread over the entire measurement period. Another study combines measurements and numerical modelling and shows that higher melt rates must have occurred in the past than those that were measured. These would lead to the closure of the channel within 250 years. Thus, neither the channel itself nor the present day melt rates endanger the stability of one of the largest Antarctic ice shelves at present.