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A main challenge for tasks on panorama lies in the distortion of objects among images. In this work, we propose a Distortion-Aware Monocular Omnidirectional (DAMO) dense depth estimation network to address this challenge on indoor panoramas with two steps. First, we introduce a distortion-aware module to extract calibrated semantic features from omnidirectional images. Specifically, we exploit deformable convolution to adjust its sampling grids to geometric variations of distorted objects on panoramas and then utilize a strip pooling module to sample against horizontal distortion introduced by inverse gnomonic projection. Second, we further introduce a plug-and-play spherical-aware weight matrix for our objective function to handle the uneven distribution of areas projected from a sphere. Experiments on the 360D dataset show that the proposed method can effectively extract semantic features from distorted panoramas and alleviate the supervision bias caused by distortion. It achieves state-of-the-art performance on the 360D dataset with high efficiency.