Cases of diabetes and related diabetic retinopathy (DR) have been increasing at an alarming rate in modern times. Early detection of DR is an important problem since it may cause permanent blindness in the late stages. In the last two decades, many different approaches have been applied in DR detection. Reviewing academic literature shows that deep neural networks (DNNs) have become the most preferred approach for DR detection. Among these DNN approaches, Convolutional Neural Network (CNN) models are the most used ones in the field of medical image classification. Designing a new CNN architecture is a tedious and time-consuming approach. Additionally, training an enormous number of parameters is also a difficult task. Due to this reason, instead of training CNNs from scratch, using pre-trained models has been suggested in recent years as transfer learning approach. Accordingly, the present study as a review focuses on DNN and Transfer Learning based applications of DR detection considering 43 publications between 2015 and 2021. The published papers are summarized using 3 figures and 10 tables, giving information about 29 pre-trained CNN models, 13 DR data sets and standard performance metrics.