Geometric primitives that consist of a group of discrete points may be viewed as one kind of abstraction and representation of lidar data at the entity level. In recent years, many efforts from different scientific communities, such as photogrammetry, computer vision and computer graphics, have been made into geometric primitive detection, regularization, and in-depth applications. The most recent lidar-based surveys focus only on reconstruction, object segmentation, and recognition, as well as the data processing techniques based on a specific platform such as mobile LiDAR. However, in this paper, lidar point clouds are understood from a new perspective, i.e., geometric primitives embedded in versatile objects in physical world. We categorize geometric primitives into two classes: shape primitives, e.g., lines, surfaces, and volumetric shapes, as well as structure primitives, represented by skeletons and edges. Interpretations of geometric primitive from multiple disciplines try to convey the significance of geometric primitives, the latest processing techniques regarding geometric primitives, and their potential possibilities in the context of lidar point clouds. To this end, applications of these primitives are reviewed with an emphasis on object extraction and reconstruction to clearly show the significances of this paper. Next, we survey and compare methods for geometric primitive extraction and then survey primitive regularization methods that add real-world geometry constrains to detected primitives. Finally, we summarize the problems and challenges and describe possible future for primitive extraction methods that can be achieved globally optimal results efficiently, even with disorganized, uneven, noisy, incomplete, and large-scale lidar point clouds.