The use of fundus images for the early screening of eye diseases is of great clinical importance. Due to its powerful performance, deep learning is becoming more and more popular in related applications, such as lesion segmentation, biomarkers segmentation, disease diagnosis and image synthesis. Therefore, it is very necessary to summarize the recent developments in deep learning for fundus images with a review paper. In this review, we introduce 143 application papers with a carefully designed hierarchy. Moreover, 33 publicly available datasets are presented. Summaries and analyses are provided for each task. Finally, limitations common to all tasks are revealed and possible solutions are given. We will also release and regularly update the state-of-the-art results and newly-released datasets at https://github.com/nkicsl/Fundus_Review to adapt to the rapid development of this field.