The amount of captured 3D data is continuously increasing, with the democratization of consumer depth cameras, the development of modern multi-view stereo capture setups and the rise of single-view 3D capture based on machine learning. The analysis and representation of this ever growing volume of 3D data, often corrupted with acquisition noise and reconstruction artifacts, is a serious challenge at the frontier between computer graphics and computer vision. To that end, segmentation and optimization are crucial analysis components of the shape abstraction process, which can themselves be greatly simplified when performed on lightened geometric formats. In this survey, we review the algorithms which extract simple geometric primitives from raw dense 3D data. After giving an introduction to these techniques, from the acquisition modality to the underlying theoretical concepts, we propose an application-oriented characterization, designed to help select an appropriate method based on one’s application needs, and compare recent approaches. We conclude by giving hints for how to evaluate these methods and a set of research challenges to be explored.