In recent years, there has been an unprecedented growth in computer vision and deep learning implementation owing to the exponential rise of computation infrastructure. The same was also reflected in retinal image analysis and successful artificial intelligence models were developed for various retinal disease diagnoses using a wide variety of visual markers obtained from eye fundus images. This article presents a comprehensive study of different deep learning strategies employed in recent times for the diagnosis of five major eye diseases, i.e., Diabetic retinopathy, Glaucoma, age-related macular degeneration, Cataract, and Retinopathy of prematurity. This article is organized according to the deep learning implementation process pipeline, where commonly used datasets, evaluation metrics, image pre-processing techniques, and deep learning backbone models are first illustrated followed by an extensive review of different strategies for each of the five mentioned retinal diseases is presented. Finally, this article summarizes eight major research directions available in the field of retinal disease diagnosis and outlines key challenges and future scope for the present research community.