Glaucoma is a leading eye disease, causing vision loss by gradually affecting peripheral vision if left untreated. Current diagnosis of glaucoma is performed by ophthalmologists, human experts who typically need to analyze different types of medical images generated by different types of medical equipment: fundus, Retinal Nerve Fiber Layer (RNFL), Optical Coherence Tomography (OCT) disc, OCT macula, perimetry, and/or perimetry deviation. Capturing and analyzing these medical images is labor intensive and time consuming. In this paper, we present a novel approach for glaucoma diagnosis and localization, only relying on fundus images that are analyzed by making use of state-of-the-art deep learning techniques. Specifically, our approach towards glaucoma diagnosis and localization leverages Convolutional Neural Networks (CNNs) and Gradient-weighted Class Activation Mapping (Grad-CAM), respectively. We built and evaluated different predictive models using a large set of fundus images, collected and labeled by ophthalmologists at Samsung Medical Center (SMC). Our experimental results demonstrate that our most effective predictive model is able to achieve a high diagnosis accuracy of 96%, as well as a high sensitivity of 96% and a high specificity of 100% for Dataset-Optic Disc (OD), a set of center-cropped fundus images highlighting the optic disc. Furthermore, we present Medinoid, a publicly-available prototype web application for computer-aided diagnosis and localization of glaucoma, integrating our most effective predictive model in its back-end.