Convolutional neural networks are widely used in computer vision applications. Although they have achieved great success, these networks can not be applied to 360 spherical images directly due to varying distortion effect. In this paper, we present distortion-aware convolutional network for spherical images. For each pixel, our network samples a non-regular grid based on its distortion level, and convolves the sampled grid using square kernels shared by all pixels. The network successively approximates large image patches from different tangent planes of viewing sphere with small local sampling grids, thus improves the computational efficiency. Our method also deals with the boundary problem, which is an inherent issue for spherical images. To evaluate our method, we apply our network in spherical image classification problems based on transformed MNIST and CIFAR-10 datasets. Compared with the baseline method, our method can get much better performance. We also analyze the variants of our network.