The precise point cloud ground segmentation is a crucial prerequisite of virtually all perception tasks for LiDAR sensors in autonomous vehicles. Especially the clustering and extraction of objects from a point cloud usually relies on an accurate removal of ground points. The correct estimation of the surrounding terrain is important for aspects of the drivability of a surface, path planning, and obstacle prediction. In this letter, we propose our system GroundGrid which relies on 2D elevation maps to solve the terrain estimation and point cloud ground segmentation problems. We evaluate the ground segmentation and terrain estimation performance of GroundGrid and compare it to other state-of-the-art methods using the SemanticKITTI dataset and a novel evaluation method relying on airborne LiDAR scanning. The results show that GroundGrid is capable of outperforming other state-of-the-art systems with an average IoU of 94.78% while maintaining a high run-time performance of 171 Hz.