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In this paper, we address panoramic semantic segmentation, which provides a full-view and dense-pixel understanding of surroundings in a holistic way. Panoramic segmentation is under-explored due to two critical challenges: (1) image distortions and object deformations on panoramas; (2) lack of annotations for training panoramic segmenters. To tackle these problems, we propose a Transformer for Panoramic Semantic Segmentation (Trans4PASS) architecture. First, to enhance distortion awareness, Trans4PASS, equipped with Deformable Patch Embedding (DPE) and Deformable MLP (DMLP) modules, is capable of handling object deformations and image distortions whenever (before or after adaptation) and wherever (shallow or deep levels) by design. We further introduce the upgraded Trans4PASS+ model, featuring DMLPv2 with parallel token mixing to improve the flexibility and generalizability in modeling discriminative cues. Second, we propose a Mutual Prototypical Adaptation (MPA) strategy for unsupervised domain adaptation. Third, aside from Pinhole-to-Panoramic (PIN2PAN) adaptation, we create a new dataset (SynPASS) with 9,080 panoramic images to explore a Synthetic-to-Real (SYN2REAL) adaptation scheme in 360 • imagery. Extensive experiments are conducted, which cover indoor and outdoor scenarios, and each of them is investigated with PIN2PAN and SYN2REAL regimens. Trans4PASS+ achieves state-of-the-art performances on four domain adaptive panoramic semantic segmentation benchmarks. Code is available at https://github.com/jamycheung/Trans4PASS.