MicroRNAs (miRNAs) are an important class of non-coding RNAs that play an essential role in the occurrence and development of various diseases. Identifying the potential miRNA-disease associations (MDAs) can be beneficial in understanding disease pathogenesis. Traditional laboratory experiments are expensive and time-consuming. Computational models have enabled systematic large-scale prediction of potential MDAs, greatly improving the research efficiency. With recent advances in deep learning, it has become an attractive and powerful technique for uncovering novel MDAs. Consequently, numerous MDA prediction methods based on deep learning have emerged. In this review, we first summarize publicly available databases related to miRNAs and diseases for MDA prediction. Next, we outline commonly used miRNA and disease similarity calculation and integration methods. Then, we comprehensively review the 48 existing deep learning-based MDA computation methods, categorizing them into classical deep learning and graph neural network-based techniques. Subsequently, we investigate the evaluation methods and metrics that are frequently used to assess MDA prediction performance. Finally, we discuss the performance trends of different computational methods, point out some problems in current research, and propose 9 potential future research directions. Data resources and recent advances in MDA prediction methods are summarized in the GitHub repository https://github.com/sheng-n/DL-miRNA-disease-association-methods.