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Robots navigating autonomously need to perceive and track the motion of objects and other agents in its surroundings. This information enables planning and executing robust and safe trajectories. To facilitate these processes, the motion should be perceived in 3D Cartesian space. However, most recent multi-object tracking (MOT) research has focused on tracking people and moving objects in 2D RGB video sequences. In this work we present JRMOT, a novel 3D MOT system that integrates information from RGB images and 3D point clouds to achieve real-time, state-of-the-art tracking performance. Our system is built with recent neural networks for re-identification, 2D and 3D detection and track description, combined into a joint probabilistic data-association framework within a multi-modal recursive Kalman architecture. As part of our work, we release the JRDB dataset, a novel large scale 2D+3D dataset and benchmark, annotated with over 2 million boxes and 3500 time consistent 2D+3D trajectories across 54 indoor and outdoor scenes. JRDB contains over 60 minutes of data including 360 degree cylindrical RGB video and 3D pointclouds in social settings that we use to develop, train and evaluate JRMOT. The presented 3D MOT system demonstrates state-of-the-art performance against competing methods on the popular 2D tracking KITTI benchmark and serves as first 3D tracking solution for our benchmark. Real-robot tests on our social robot JackRabbot indicate that the system is capable of tracking multiple pedestrians fast and reliably. We provide the ROS code of our tracker at https://sites.google.com/view/jrmot.