Earth observation technologies, such as optical imaging and synthetic aperture radar (SAR), provide excellent means to monitor ever-growing urban environments continuously. Notably, in the case of large-scale disasters (e.g., tsunamis and earthquakes), in which a response is highly time-critical, images from both data modalities can complement each other to accurately convey the full damage condition in the disaster's aftermath. However, due to several factors, such as weather and satellite coverage, it is often uncertain which data modality will be the first available for rapid disaster response efforts. Hence, novel methodologies that can utilize all accessible EO datasets are essential for disaster management. In this study, we have developed a global multisensor and multitemporal dataset for building damage mapping. We included building damage characteristics from three disaster types, namely, earthquakes, tsunamis, and typhoons, and considered three building damage categories. The global dataset contains high-resolution optical imagery and high-to-moderate-resolution multiband SAR data acquired before and after each disaster. Using this comprehensive dataset, we analyzed five data modality scenarios for damage mapping: single-mode (optical and SAR datasets), cross-modal (pre-disaster optical and post-disaster SAR datasets), and mode fusion scenarios. We defined a damage mapping framework for the semantic segmentation of damaged buildings based on a deep convolutional neural network algorithm. We compare our approach to another state-of-the-art baseline model for damage mapping. The results indicated that our dataset, together with a deep learning network, enabled acceptable predictions for all the data modality scenarios.