The continuous and precise mapping of glacier calving fronts is essential for monitoring and understanding rapid glacier changes in Antarctica and Greenland, which have the potential for significant sea level rise within the current century. This effort has been mostly restricted to the slow and painstaking manual digitalization of the calving front positions in thousands of satellite imagery products. Here, we have developed a machine learning toolkit to robustly and automatically detect glacier calving front margins in satellite imagery. The toolkit is based on semantic image segmentation using Convolutional Neural Networks (CNN) with a modified U-Net architecture to isolate the calving fronts from satellite images after having been trained with a dataset of images and their corresponding manually-determined calving fronts. As a case study we train our neural network on a varied set Landsat images with lowered resolutions from Jakobshavn, Sverdrup, and Kangerlussuaq glaciers, Greenland and test the results on novel images from Helheim glacier, Greenland to evaluate the performance of the approach. The neural network is able to identify the calving front in new images with a mean deviation of 96.3 m from the true fronts, equivalent to 1.97 pixels on average, while the corresponding error for manually-determined fronts on the same resolution images is 92.5 m. We find that the trained neural network significantly outperforms common edge detection techniques, and can be used to continuously map out calving-ice fronts with a variety of data products.