A recent approach for object detection and human pose estimation is to regress bounding boxes or human keypoints from a central point on the object or person. While this center-point regression is simple and efficient, we argue that the image features extracted at a central point contain limited information for predicting distant keypoints or bounding box boundaries, due to object deformation and scale/orientation variation. To facilitate inference, we propose to instead perform regression from a set of points placed at more advantageous positions. This point set is arranged to reflect a good initialization for the given task, such as modes in the training data for pose estimation, which lie closer to the ground truth than the central point and provide more informative features for regression. As the utility of a point set depends on how well its scale, aspect ratio and rotation matches the target, we adopt the anchor box technique of sampling these transformations to generate additional point-set candidates. We apply this proposed framework, called Point-Set Anchors, to object detection, instance segmentation, and human pose estimation. Our results show that this general-purpose approach can achieve performance competitive with state-of-the-art methods for each of these tasks. Code is available at url{https://github.com/FangyunWei/PointSetAnchor}