Diffusion Magnetic Resonance Imaging (dMRI) is an imaging technique which enables analysis of the brain tissue at a microscopic scale, particularly the analysis of white matter. Given a high enough angular resolution, a common way to explain the measured signal is via fiber orientation distribution function (fODF). This function describes the orientation and volume fraction of axon bundles within each voxel and is an essential ingredient of tractography. In this work, we have investigated a deep learning approach for the fODF estimation. U-nets enable fast and high resolution inference by combining multi-scale features from contracting and expanding parts of the network. As dMRI signals are most commonly acquired on spheres, we propose a spherical U-net which is adjusted to the properties of the dMRI data, namely its real nature, antipodal symmetry, uniform sampling and axial symmetry of the signals corresponding to individual fibers. We compared our model with another deep learning approach based on a 3D convolutional neural network and a state-of-the-art approach-multi-shell multi-tissue constrained spherical deconvolution, on real data from Human Connectome Project and synthetic data generated using ball and stick model. The methods are compared in terms of mean square error and mean angular error for dMRI signals of different angular resolutions. Provided quantitative analyses show improved performance with our approach even with significantly reduced number of parameters and results obtained on synthetic data indicate its robustness with respect to noise. Qualitative results illustrating the performance of the methods are also presented.