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The panorama image can simultaneously demonstrate complete information of the surrounding environment and has many advantages in virtual tourism, games, robotics, etc. However, the progress of panorama depth estimation cannot completely solve the problems of distortion and discontinuity caused by the commonly used projection methods. This paper proposes SphereDepth, a novel panorama depth estimation method that predicts the depth directly on the spherical mesh without projection preprocessing. The core idea is to establish the relationship between the panorama image and the spherical mesh and then use a deep neural network to extract features on the spherical domain to predict depth. To address the efficiency challenges brought by the high-resolution panorama data, we introduce two hyper-parameters for the proposed spherical mesh processing framework to balance the inference speed and accuracy. Validated on three public panorama datasets, SphereDepth achieves comparable results with the state-of-the-art methods of panorama depth estimation. Benefiting from the spherical domain setting, SphereDepth can generate a high-quality point cloud and significantly alleviate the issues of distortion and discontinuity.