The rapid development of remote sensing technologies and the availability of many satellite and aerial sensors have boosted the collection of large volumes of high-resolution images, promoting progress in a wide range of applications. As a consequence, Object detection (OD) in aerial images has gained much interest in the last few years. However, the development of object detectors requires a massive amount of carefully labeled data. Since annotating datasets is very time-consuming and may require expert knowledge, a consistent number of weakly supervised object localization (WSOL) and detection (WSOD) methods have been developed. These approaches exploit only coarse-grained metadata, typically whole image labels, to train object detectors. However, many challenges remain open due to the missing location information in the training process of WSOD approaches and to the complexity of remote sensing images. Furthermore, methods studied for natural images may not be directly applicable to remote sensing images (RSI) and may require carefully designed adaptations. This work provides a comprehensive survey of the recent achievements of remote sensing weakly supervised object detection (RSWSOD). An analysis of the challenges related to RSWSOD is presented, the advanced techniques developed to improve WSOD are summarized, the available benchmarking datasets are described and a discussion of future directions of RSWSOD research is provided.