The ability to detect and segment moving objects in a scene is essential for building consistent maps, making future state predictions, avoiding collisions, and planning. In this paper, we address the problem of moving object segmentation from 3D LiDAR scans. We propose a novel approach that pushes the current state of the art in LiDAR-only moving object segmentation forward to provide relevant information for autonomous robots and other vehicles. Instead of segmenting the point cloud semantically, i.e., predicting the semantic classes such as vehicles, pedestrians, roads, etc., our approach accurately segments the scene into moving and static objects, i.e., also distinguishing between moving cars vs. parked cars. Our proposed approach exploits sequential range images from a rotating 3D LiDAR sensor as an intermediate representation combined with a convolutional neural network and runs faster than the frame rate of the sensor. We compare our approach to several other state-of-the-art methods showing superior segmentation quality in urban environments. Additionally, we created a new benchmark for LiDAR-based moving object segmentation based on SemanticKITTI. We published it to allow other researchers to compare their approaches transparently and we furthermore published our code.