Glaucoma is a prevalent cause of blindness worldwide. If not treated promptly, it can cause vision and quality of life to deteriorate. According to statistics, glaucoma affects approximately 65 million individuals globally. Fundus image segmentation depends on the optic disc (OD) and optic cup (OC). This paper proposes a computational model to segment and classify retinal fundus images for glaucoma detection. Different data augmentation techniques were applied to prevent overfitting while employing several data pre-processing approaches to improve the image quality and achieve high accuracy. The segmentation models are based on an attention U-Net with three separate convolutional neural networks (CNNs) backbones: Inception-v3, visual geometry group 19 (VGG19), and residual neural network 50 (ResNet50). The classification models also employ a modified version of the above three CNN architectures. Using the RIM-ONE dataset, the attention U-Net with the ResNet50 model as the encoder backbone, achieved the best accuracy of 99.58% in segmenting OD. The Inception-v3 model had the highest accuracy of 98.79% for glaucoma classification among the evaluated segmentation, followed by the modified classification architectures.