Deep learning has dramatically improved object recognition, speech recognition, medical image analysis and many other fields. Optical coherence tomography (OCT) has become a standard of care imaging modality for ophthalmology. We asked whether deep learning could be used to segment cornea OCT images. Using a custom-built ultrahigh-resolution OCT system, we scanned 72 healthy eyes and 70 keratoconic eyes. In total, 20,160 images were labeled and used for the training in a supervised learning approach. A custom neural network architecture called CorneaNet was designed and trained. Our results show that CorneaNet is able to segment both healthy and keratoconus images with high accuracy (validation accuracy: 99.56%). Thickness maps of the three main corneal layers (epithelium, Bowman's layer and stroma) were generated both in healthy subjects and subjects suffering from keratoconus. CorneaNet is more than 50 times faster than our previous algorithm. Our results show that deep learning algorithms can be used for OCT image segmentation and could be applied in various clinical settings. In particular, CorneaNet could be used for early detection of keratoconus and more generally to study other diseases altering corneal morphology.