Intelligent vehicles clearly benefit from the expanded Field of View (FoV) of the 360-degree sensors, but the vast majority of available semantic segmentation training images are captured with pinhole cameras. In this work, we look at this problem through the lens of domain adaptation and bring panoramic semantic segmentation to a setting, where labelled training data originates from a different distribution of conventional pinhole camera images. First, we formalize the task of unsupervised domain adaptation for panoramic semantic segmentation, where a network trained on labelled examples from the source domain of pinhole camera data is deployed in a different target domain of panoramic images, for which no labels are available. To validate this idea, we collect and publicly release DensePASS - a novel densely annotated dataset for panoramic segmentation under cross-domain conditions, specifically built to study the Pinhole-to-Panoramic transfer and accompanied with pinhole camera training examples obtained from Cityscapes. DensePASS covers both, labelled- and unlabelled 360-degree images, with the labelled data comprising 19 classes which explicitly fit the categories available in the source domain (i.e. pinhole) data. To meet the challenge of domain shift, we leverage the current progress of attention-based mechanisms and build a generic framework for cross-domain panoramic semantic segmentation based on different variants of attention-augmented domain adaptation modules. Our framework facilitates information exchange at local- and global levels when learning the domain correspondences and improves the domain adaptation performance of two standard segmentation networks by 6.05% and 11.26% in Mean IoU.