To review the existing deep learning applications for diagnosing diabetic retinopathy and retinopathy of prematurity diseases, the available public retinal databases for the diseases and apply the International Journal of Medical Informatics (IJMEDI) checklist were assessed the quality of included studies; an in-depth literature search in Scopus, Web of Science, IEEE and ACM databases targeting articles from inception up to 31st January 2023 was done by two independent reviewers. In the review, 26 out of 1476 articles with a total of 36 models were included. Data size and model validation were found to be challenges for most studies. Deep learning models are gaining focus in the development of medical diagnosis tools and applications. However, there seems to be a critical issue with most of the studies being published, with some not including information about data sources and data sizes which is important for their performance verification.