Glaucoma is a chronic eye disease that may lead to permanent vision loss if it is not diagnosed and treated at an early stage. The disease originates from an irregular behavior in the drainage flow of the eye that eventually leads to an increase in intraocular pressure, which in the severe stage of the disease deteriorates the optic nerve head and leads to vision loss. Medical follow-ups to observe the retinal area are needed periodically by ophthalmologists, who require an extensive degree of skill and experience to interpret the results appropriately. To improve on this issue, algorithms based on deep learning techniques have been designed to screen and diagnose glaucoma based on retinal fundus image input and to analyze images of the optic nerve and retinal structures. Therefore, the objective of this paper is to provide a systematic analysis of 52 state-of-the-art relevant studies on the screening and diagnosis of glaucoma, which include a particular dataset used in the development of the algorithms, performance metrics, and modalities employed in each article. Furthermore, this review analyzes and evaluates the used methods and compares their strengths and weaknesses in an organized manner. It also explored a wide range of diagnostic procedures, such as image pre-processing, localization, classification, and segmentation. In conclusion, automated glaucoma diagnosis has shown considerable promise when deep learning algorithms are applied. Such algorithms could increase the accuracy and efficiency of glaucoma diagnosis in a better and faster manner.