In human brain MRI studies, it is of great importance to accurately parcellate cortical surfaces into anatomically and functionally meaningful regions. In this paper, we propose a novel end-to-end deep learning method by formulating surface parcellation as a semantic segmentation task on the sphere. To extend the convolutional neural networks (CNNs) to the spherical space, corresponding operations of surface convolution, pooling and upsampling are first developed to deal with data representation on spherical surface meshes, and then spherical CNNs are constructed accordingly. Specifically, the U-Net and SegNet architectures are transformed to the spherical representation for neonatal cortical surface parcellation. Experimental results on 90 neonates indicate the effectiveness and efficiency of our proposed spherical U-Net, in comparison with the spherical SegNet and the previous patch-wise classification method.