Hyperspectral image (HSI) denoising is a crucial preprocessing procedure to improve the performance of the subsequent HSI interpretation and applications. In this paper, a novel deep learning-based method for this task is proposed, by learning a nonlinear end-to-end mapping between the noisy and clean HSIs with a combined spatial-spectral deep convolutional neural network (HSID-CNN). Both the spatial and spectral information are simultaneously assigned to the proposed network. In addition, multiscale feature extraction and multilevel feature representation are, respectively, employed to capture both the multiscale spatial-spectral feature and fuse different feature representations for the final restoration. The simulated and real-data experiments demonstrate that the proposed HSID-CNN outperforms many of the mainstream methods in both the quantitative evaluation indexes, visual effects, and HSI classification accuracy.