Accurate calculation of drug-target affinity (DTA) is crucial for various applications in the pharmaceutical industry, including drug screening, design, and repurposing. However, traditional machine learning methods for calculating DTA often lack accuracy, posing a significant challenge in accurately predicting DTA. Fortunately, deep learning has emerged as a promising approach in computational biology, leading to the development of various deep learning-based methods for DTA prediction. To support researchers in developing novel and highly precision methods, we have provided a comprehensive review of recent advances in predicting DTA using deep learning. We firstly conducted a statistical analysis of commonly used public datasets, providing essential information and introducing the used fields of these datasets. We further explored the common representations of sequences and structures of drugs and targets. These analyses served as the foundation for constructing DTA prediction methods based on deep learning. Next, we focused on explaining how deep learning models, such as Convolutional Neural Networks (CNNs), Recurrent Neural Networks (RNNs), Transformer, and Graph Neural Networks (GNNs), were effectively employed in specific DTA prediction methods. We highlighted the unique advantages and applications of these models in the context of DTA prediction. Finally, we conducted a performance analysis of multiple state-of-the-art methods for predicting DTA based on deep learning. The comprehensive review aimed to help researchers understand the shortcomings and advantages of existing methods, and further develop high-precision DTA prediction tool to promote the development of drug discovery.