Glaucoma is a degenerative eye disease that affects the optic nerve. If untreated, it can lead to irreversible vision loss and blindness. Early detection and treatment of glaucoma are essential to prevent and control irreversible vision loss. In this paper, we have proposed a deep learning-based method for the automated detection of glaucoma from fundus images. We have designed and implemented two convolutional neural network models, namely modified VGG16 and modified ResNet-50, for automatic feature extraction and classification. On the ACRIMA dataset, the proposed modified VGG16 achieved 94% accuracy, 80.95% specificity and 97.47% sensitivity. In comparison, the modified ResNet-50 model achieved 93% accuracy, 85.71% specificity and 94.94% sensitivity. Both the models outperformed the existing glaucoma detection methods in literature and provided state-of-the-art results. The proposed deep learning models have the potential to significantly improve the accuracy, speed, and convenience of glaucoma screening and diagnosis, especially in resource-limited settings. The results of our study suggest that deep learning models can serve as practical tools for automated glaucoma detection and assist clinicians in early diagnosis, leading to timely treatment.