Panoramic imaging enables capturing 360° images with an ultra-wide Field-of-View (FoV) for dense omnidirectional perception, which is critical to applications, such as autonomous driving and augmented reality, etc. However, current panoramic semantic segmentation methods fail to identify outliers, and pinhole Out-of-distribution Segmentation (OoS) models perform unsatisfactorily in the panoramic domain due to pixel distortions and background clutter. To address these issues, we introduce a new task, Panoramic Out-of-distribution Segmentation (PanOoS), with the aim of achieving comprehensive and safe scene understanding. Furthermore, we propose the first solution, POS, which adapts to the characteristics of panoramic images through text-guided prompt distribution learning. Specifically, POS integrates a disentanglement strategy designed to materialize the cross-domain generalization capability of CLIP. The proposed Prompt-based Restoration Attention (PRA) optimizes semantic decoding by prompt guidance and self-adaptive correction, while Bilevel Prompt Distribution Learning (BPDL) refines the manifold of per-pixel mask embeddings via semantic prototype supervision. Besides, to compensate for the scarcity of PanOoS datasets, we establish two benchmarks: DenseOoS, which features diverse outliers in complex environments, and QuadOoS, captured by a quadruped robot with a panoramic annular lens system. Extensive experiments demonstrate superior performance of POS, with AuPRC improving by 34.25% and FPR95 decreasing by 21.42% on DenseOoS, outperforming state-of-the-art pinhole-OoS methods. Moreover, POS achieves leading closed-set segmentation capabilities and advances the development of panoramic understanding. Code and datasets will be available at https://github.com/MengfeiD/PanOoS.