Glaucoma is a major eye disease, leading to vision loss in the absence of proper medical treatment. Current diagnosis of glaucoma is performed by ophthalmologists who are often analyzing several types of medical images generated by different types of medical equipment. Capturing and analyzing these medical images is labor-intensive and expensive. In this paper, we present a novel computational approach towards glaucoma diagnosis and localization, only making use of eye fundus images that are analyzed by state-of-the-art deep learning techniques. Specifically, our approach leverages Convolutional Neural Networks (CNNs) and Gradient-weighted Class Activation Mapping (Grad-CAM) for glaucoma diagnosis and localization, respectively. Quantitative and qualitative results, as obtained for a small-sized dataset with no segmentation ground truth, demonstrate that the proposed approach is promising, for instance achieving an accuracy of 0.91$pm0.02$ and an ROC-AUC score of 0.94 for the diagnosis task. Furthermore, we present a publicly available prototype web application that integrates our predictive model, with the goal of making effective glaucoma diagnosis available to a wide audience.