Over the past years, use of the retinal fundus images has increased for diagnosis of retinal diseases. Glaucoma is a disease which causes damage to the optic nerve of the eye resulting in deteriorated vision. Once diagnosed, the disease cannot be treated completely, but timely detection can further control the effect of glaucoma. Detection is usually performed by analyzing the optic disc followed by optic cup present on an exit of ganglion cells in the eye. Using retinal fundus images and image processing approaches, various research studies have been published till date, but the problem of accurate segmentation of disc and cup is still a major concern. This paper aims to analyze various segmentation approaches used by different researchers for optic disc followed by the optic cup and its classification for diagnosis of glaucoma. Also, the paper addresses various research gaps and challenges which need to be dealt with for improving the accuracy of segmentation and classification.