Histopathological images stained with hematoxylin and eosin (H&E) are crucial for cancer diagnosis and prognosis. However, color variations caused by differences in tissue preparation and scanning devices can lead to data distribution discrepancies, adversely affecting the performance of downstream algorithms in tasks like classification, segmentation, and detection. To address these issues, stain normalization methods have been developed to standardize color distributions across images from various sources. Recent advancements in deep learning-based stain normalization methods have shown significant promise due to their minimal preprocessing requirements, independence from reference templates, and robustness. This review examines 115 publications to explore the latest developments in this field. We first outline the evaluation metrics and publicly available datasets used for assessing stain normalization methods. Next, we systematically review deep learning-based approaches, including supervised, unsupervised, and self-supervised methods, categorizing them by core technologies and analyzing their contributions and limitations. Finally, we discuss current challenges and future directions, aiming to provide researchers with a comprehensive understanding of the field, promote further development, and accelerate the progress of intelligent cancer diagnosis.