Objectives: To evaluate the performance of convolutional neural network (CNN) architectures to distinguish eyes with glaucoma from normal eyes. Materials and Methods: A total of 9,950 fundus photographs of 5,388 patients from the database of Eskişehir Osmangazi University Faculty of Medicine Ophthalmology Clinic were labelled as glaucoma, glaucoma suspect, or normal by three different experienced ophthalmologists. The categorized fundus photographs were evaluated using a state-of-the-art two-dimensional CNN and compared with deep residual networks (ResNet) and very deep neural networks (VGG). The accuracy, sensitivity, and specificity of glaucoma detection with the different algorithms were evaluated using a dataset of 238 normal and 320 glaucomatous fundus photographs. For the detection of suspected glaucoma, ResNet-101 architectures were tested with a data set of 170 normal, 170 glaucoma, and 167 glaucoma-suspect fundus photographs. Results: Accuracy, sensitivity, and specificity in detecting glaucoma were 96.2%, 99.5%, and 93.7% with ResNet-50; 97.4%, 97.8%, and 97.1% with ResNet-101; 98.9%, 100%, and 98.1% with VGG-19, and 99.4%, 100%, and 99% with the 2D CNN, respectively. Accuracy, sensitivity, and specificity values in distinguishing glaucoma suspects from normal eyes were 62%, 68%, and 56% and those for differentiating glaucoma from suspected glaucoma were 92%, 81%, and 97%, respectively. While 55 photographs could be evaluated in 2 seconds with CNN, a clinician spent an average of 24.2 seconds to evaluate a single photograph. Conclusion: An appropriately designed and trained CNN was able to distinguish glaucoma with high accuracy even with a small number of fundus photographs.