Seasonal snow is an essential water resource in many mountain regions. However, the spatio-temporal variability in mountain snow depth or snow water equivalent (SWE) at regional to global scales is not well understood due to the lack of high-resolution satellite observations and robust retrieval algorithms. We investigate the ability of the Sentinel-1 mission to monitor snow depth at sub-kilometer (100m, 500m, and 1km) resolutions over the European Alps for 2017-2019. The Sentinel-1 backscatter observations, especially in cross-polarization, show a high correlation with regional model simulations of snow depth over Austria and Switzerland. The observed changes in radar backscatter with the accumulation or ablation of snow are used in an empirical change detection algorithm to retrieve snow depth. The algorithm includes the detection of dry and wet snow conditions. Compared to in situ measurements at 743 sites in the European Alps, dry snow depth retrievals at 500m and 1km resolution have a spatio-temporal correlation of 0.89. The mean absolute error equals 20%-30% of the measured values for snow depths between 1.5 and 3m. The performance slightly degrades for retrievals at the finer 100m spatial resolution as well as for retrievals of shallower and deeper snow. The results demonstrate the ability of Sentinel-1 to provide snow estimates in mountainous regions where satellite-based estimates of snow mass are currently lacking. The retrievals can improve our knowledge of seasonal snow mass in areas with complex topography and benefit a number of applications, such as water resource management, flood forecasting, and numerical weather prediction. However, future research is recommended to further investigate the physical basis of the sensitivity of Sentinel-1 backscatter observations to snow accumulation.