This paper presents Point Convolutional Neural Networks (PCNN): a novelframework for applying convolutional neural networks to point clouds. Theframework consists of two operators: extension and restriction, mapping pointcloud functions to volumetric functions and vise-versa. A point cloudconvolution is defined by pull-back of the Euclidean volumetric convolution viaan extension-restriction mechanism. The point cloud convolution is computationally efficient, invariant to theorder of points in the point cloud, robust to different samplings and varyingdensities, and translation invariant, that is the same convolution kernel isused at all points. PCNN generalizes image CNNs and allows readily adaptingtheir architectures to the point cloud setting. Evaluation of PCNN on three central point cloud learning benchmarksconvincingly outperform competing point cloud learning methods, and the vastmajority of methods working with more informative shape representations such assurfaces and/or normals.