Corneal pathologies are leading causes of blindness and represent a world health problem according to the world health organization. Early detection of corneal diseases is necessary to prevent blindness. In this paper, we use transfer learning with pretrained deep learning networks to diagnose three common corneal diseases, namely, dry eye, Fuchs' endothelial dystrophy, and keratoconus as well as healthy eyes using only optical coherence tomography (OCT) images. Corneal OCT scans were obtained from 413 eyes of 269 patients and used to train, validate, and test the networks. All networks achieved all-category accuracy values > 99%, categorical area under curve values > 0:99, categorical specificity values > 99%, and categorical sensitivity values > 99% on the training, validation, and testing, respectively. The work in this paper has clinical significance and can potentially be applied in clinical practice to potentially solve a significant world health problem.