PointNet has revolutionized how we think about representing point clouds. For classification and segmentation tasks, the approach and its subsequent variants/extensions are considered state-of-the-art. To date, the successful application of PointNet to point cloud registration has remained elusive. In this paper we argue that PointNet itself can be thought of as a learnable" imaging" function. As a consequence, classical vision algorithms for image alignment can be brought to bear on the problem--namely the Lucas & Kanade (LK) algorithm. Our central innovations stem from:(i) how to modify the LK algorithm to accommodate the PointNet imaging function, and (ii) unrolling PointNet and the LK algorithm into a single trainable recurrent deep neural network. We describe the architecture, and compare its performance against state-of-the-art in several common registration scenarios. The architecture offers some remarkable properties including: generalization across shape categories and computational efficiency--opening up new paths of exploration for the application of deep learning to point cloud registration. Code and videos are available at https://github. com/hmgoforth/PointNetLK.