3D object recognition in scenes with occlusion and clutter is a difficult task. In this paper, we introduce a method that exploits the geometric scale-variability to aid in this task. Our key insight is to leverage the rich discriminative information provided by the scale variation of local geometric structures to constrain the massive search space of potential correspondences between model and scene points. In particular, we exploit the geometric scale variability in the form of the intrinsic geometric scale of each computed feature, the hierarchy induced within the set of these intrinsic geometric scales, and the discriminative power of the local scale-dependent/invariant 3D shape descriptors. The method exploits the added information in a hierarchical coarse-to-fine manner that lets it cull the space of all potential correspondences effectively. We experimentally evaluate the accuracy of our method on an extensive set of real scenes with varying amounts of partial occlusion and achieve recognition rates higher than the state-of-the-art. Furthermore, for the first time we systematically demonstrate the method's ability to accurately localize objects despite changes in their global scales.