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Combining Simultaneous Localisation and Mapping (SLAM) estimation and dynamic scene modelling can highly benefit robot autonomy in dynamic environments. Robot path planning and obstacle avoidance tasks rely on accurate estimations of the motion of dynamic objects in the scene. This paper presents VDO-SLAM, a robust visual dynamic object-aware SLAM system that exploits semantic information to enable accurate motion estimation and tracking of dynamic rigid objects in the scene without any prior knowledge of the objects' shape or geometric models. The proposed approach identifies and tracks the dynamic objects and the static structure in the environment and integrates this information into a unified SLAM framework. This results in highly accurate estimates of the robot's trajectory and the full SE(3) motion of the objects as well as a spatiotemporal map of the environment. The system is able to extract linear velocity estimates from objects' SE(3) motion providing an important functionality for navigation in complex dynamic environments. We demonstrate the performance of the proposed system on a number of real indoor and outdoor datasets and the results show consistent and substantial improvements over the state-of-the-art algorithms. An open-source version of the source code is available.