Deep learning (DL) enables the creation of computational models comprising multiple processing layers that learn data representations at multiple levels of abstraction. In the recent past, the use of deep learning has been proliferating, yielding promising results in applications across a growing number of fields, most notably in image processing, medical image analysis, data analysis, and bioinformatics. DL algorithms have also had a significant positive impact through yielding improvements in screening, recognition, segmentation, prediction, and classification applications across different domains of healthcare, such as those concerning the abdomen, cardiac, pathology, and retina. Given the extensive body of recent scientific contributions in this discipline, a comprehensive review of deep learning developments in the domain of diabetic retinopathy (DR) analysis, viz., screening, segmentation, prediction, classification, and validation, is presented here. A critical analysis of the relevant reported techniques is carried out, and the associated advantages and limitations highlighted, culminating in the identification of research gaps and future challenges that help to inform the research community to develop more efficient, robust, and accurate DL models for the various challenges in the monitoring and diagnosis of DR.