Most successful object recognition systems rely on binary classification, deciding only if an object is present or not, but not providing information on the actual object location. To estimate the object's location, one can take a sliding window approach, but this strongly increases the computational cost because the classifier or similarity function has to be evaluated over a large set of candidate subwindows. In this paper, we propose a simple yet powerful branch and bound scheme that allows efficient maximization of a large class of quality functions over all possible subimages. It converges to a globally optimal solution typically in linear or even sublinear time, in contrast to the quadratic scaling of exhaustive or sliding window search. We show how our method is applicable to different object detection and image retrieval scenarios. The achieved speedup allows the use of classifiers for localization that formerly were considered too slow for this task, such as SVMs with a spatial pyramid kernel or nearest-neighbor classifiers based on the lambda2 distance. We demonstrate state-of-the-art localization performance of the resulting systems on the UIUC Cars data set, the PASCAL VOC 2006 data set, and in the PASCAL VOC 2007 competition.