Abstract: We propose new, and robust, loss functions for the point cloud registration problem. Our loss functions are inspired by the Best Buddies Similarity (BBS) measure that counts the number of mutual nearest neighbors between two point sets. This measure has been shown to be robust to outliers and missing data in the case of template matching for images. We present several algorithms, collectively named \emph{Best Buddy Registration (BBR)}, where each algorithm consists of optimizing one of these loss functions with Adam gradient descent. The loss functions differ in several ways, including the distance function used (point-to-point vs. point-to-plane), and how the BBS measure is combined with the actual distances between pairs of points. Experiments on various data sets, both synthetic and real, demonstrate the effectiveness of the BBR algorithms, showing that they are quite robust to noise, outliers, and distractors, and cope well with extremely sparse point clouds. One variant, BBR-F, achieves state-of-the-art accuracy in the registration of automotive lidar scans taken up to several seconds apart, from the KITTI and Apollo-Southbay datasets.