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This paper presents a deep normal filtering network, called DNF-Net, for mesh denoising. To better capture local geometry, our network processes the mesh in terms of local patches extracted from the mesh. Overall, DNF-Net is an end-to-end network that takes patches of facet normals as inputs and directly outputs the corresponding denoised facet normals of the patches. In this way, we can reconstruct the geometry from the denoised normals with feature preservation. Besides the overall network architecture, our contributions include a novel multi-scale feature embedding unit, a residual learning strategy to remove noise, and a deeply-supervised joint loss function. Compared with the recent data-driven works on mesh denoising, DNF-Net does not require manual input to extract features and better utilizes the training data to enhance its denoising performance. Finally, we present comprehensive experiments to evaluate our method and demonstrate its superiority over the state of the art on both synthetic and real-scanned meshes.