Glaucoma is a heterogeneous group of diseases that are characterized by loss of retinal ganglion cells, which damages the optic nerve head (ONH) and visual field. If glaucoma, the most frequent cause of irretrievable vision loss, is detected at an initial stage, the rate of blindness may be reduced by nearly 50%-55%. Manual diagnosis is a laborious task; it is fairly time consuming and requires a skilled medical provider. With the lack of trained professionals in developing countries, automatic glaucoma diagnosis becomes an increasingly vital tool that aids in detection and disease risk analysis. Analyses of the optic disc (OD) and optic cup (OC) are normally performed to assess ONH damage. But of the numerous reported research reports that show results using machine-learning and image-processing approaches, major concern lies in the accuracy of segmenting and classifying OD and OC. The objective of the current study is to outline state-of-the-art image-processing techniques that are used to detect glaucoma early via segmenting and OD and OC classification. We also present research findings and limitations thereof that must be addressed to achieve higher accuracy to improve segmentation and classification quality.