Advancements in synthetic aperture radar (SAR) technology have enabled large-scale observations and high-resolution imaging. Consequently, SAR images now contain numerous small-sized objects with weak features, including aircraft, vehicles, tanks, and ships, which are of high value in civilian and key military assets. However, accurately detecting these objects poses a significant challenge due to their small size, dense connectivity, and variable morphology. Deep learning technology has ushered in a new era of progress in SAR object detection. Researchers have made substantial strides by fine-tuning and optimizing deep learning networks to address the imaging characteristics and detection challenges associated with weak SAR objects. This study provides a comprehensive review of deep learning-based methodologies for weak object detection in SAR images. The primary focus is on datasets and methods, providing a thorough analysis of the principal challenges encountered in SAR weak object detection. This study also summarizes the characteristics and application scenarios of recent detection methods, as well as collates and organizes publicly available datasets and common performance evaluation metrics. In conclusion, this study provides an overview of the current application status of SAR weak object detection and offers insights into future development trends.