Domain adaptation for classification has achieved significant progress in natural images but not in remote-sensing images due to huge differences in data-imaging mechanisms between different modalities and inconsistencies in class labels among existing datasets. More importantly, the lack of cross-domain benchmark datasets has become a major obstacle to the development of scene classification in multimodal remote-sensing images. In this paper, we present a cross-domain dataset of multimodal remote-sensing scene classification (MRSSC). The proposed MRSSC dataset contains 26,710 images of 7 typical scene categories with 4 distinct domains that are collected from Tiangong-2, a Chinese manned spacecraft. Based on this dataset, we evaluate several representative domain adaptation algorithms on three cross-domain tasks to build baselines for future research. The results demonstrate that the domain adaptation algorithm can reduce the differences in data distribution between different domains and improve the accuracy of the three tasks to varying degrees. Furthermore, MRSSC also achieved fairly results in three applications: cross-domain data annotation, weakly supervised object detection and data retrieval. This dataset is believed to stimulate innovative research ideas and methods in remote-sensing cross-domain scene classification and remote-sensing intelligent interpretation.