Bounding-box regression is a popular technique to refine or predict localization boxes in recent object detection approaches. Typically, bounding-box regressors are trained to regress from either region proposals or fixed anchor boxes to nearby bounding boxes of a pre-defined target object classes. This paper investigates whether the technique is generalizable to unseen classes and is transferable to other tasks beyond supervised object detection. To this end, we propose a class-agnostic and anchor-free box regressor, dubbed Universal Bounding-Box Regressor (UBBR), which predicts a bounding box of the nearest object from any given box. Trained on a relatively small set of annotated images, UBBR successfully generalizes to unseen classes, and can be used to improve localization in many vision problems. We demonstrate its effectivenss on weakly supervised object detection and object discovery.