Motivation Effective use of evolutionary information has recently led to tremendous progress in computational prediction of three-dimensional (3D) structures of proteins and their complexes. Despite the progress, the accuracy of predicted structures tends to vary considerably from case to case. Since the utility of computational models depends on their accuracy, reliable estimates of deviation between predicted and native structures are of utmost importance. Results For the first time, we present a deep convolutional neural network (CNN) constructed on a Voronoi tessellation of 3D molecular structures. Despite the irregular data domain, our data representation allows us to efficiently introduce both convolution and pooling operations and train the network in an end-to-end fashion without precomputed descriptors. The resultant model, VoroCNN, predicts local qualities of 3D protein folds. The prediction results are competitive to state of the art and superior to the previous 3D CNN architectures built for the same task. We also discuss practical applications of VoroCNN, for example, in recognition of protein binding interfaces. Availability The model, data, and evaluation tests are available at https://team.inria.fr/nano-d/software/vorocnn/. Supplementary information Supplementary data are available at Bioinformatics online.