Parsing an unstructured point set into constituent local geometry structures (e.g., edges or surfaces) would be helpful for understanding and representing point clouds. This motivates us to design a deep architecture to model the hierarchical geometry from points, edges, surfaces (triangles), to super-surfaces (adjacent surfaces) for the thorough analysis of point clouds. In this paper, we present a novel Hierarchical Geometry Network (HGNet) that integrates such hierarchical geometry structures from super-surfaces, surfaces, edges, to points in a top-down manner for learning point cloud representations. Technically, we first construct the edges between every two neighbor points. A point-level representation is learnt with edge-to-point aggregation, i.e., aggregating all connected edges into the anchor point. Next, as every two neighbor edges compose a surface, we obtain the edge-level representation of each anchor edge via surface-to-edge aggregation over all neighbor surfaces. Furthermore, the surface-level representation is achieved through super-surface-to-surface aggregation by transforming all super-surfaces into the anchor surface. A Transformer structure is finally devised to unify all the point-level, edge-level, and surface-level features into the holistic point cloud representations. Extensive experiments on four point cloud analysis datasets demonstrate the superiority of HGNet for 3D object classification and part/semantic segmentation tasks. More remarkably, HGNet achieves the overall accuracy of 89.2% on ScanObjectNN, improving PointNeXt-S by 1.5%.