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3D single object tracking plays an essential role in many applications, such as autonomous driving. It remains a challenging problem due to the large appearance variation and the sparsity of points caused by occlusion and limited sensor capabilities. Therefore, contextual information across two consecutive frames is crucial for effective object tracking. However, points containing such useful information are often overlooked and cropped out in existing methods, leading to insufficient use of important contextual knowledge. To address this issue, we propose CXTrack, a novel transformer-based network for 3D object tracking, which exploits ConteXtual information to improve the tracking results. Specifically, we design a target-centric transformer network that directly takes point features from two consecutive frames and the previous bounding box as input to explore contextual information and implicitly propagate target cues. To achieve accurate localization for objects of all sizes, we propose a transformer-based localization head with a novel center embedding module to distinguish the target from distractors. Extensive experiments on three large-scale datasets, KITTI, nuScenes and Waymo Open Dataset, show that CXTrack achieves state-of-the-art tracking performance while running at 34 FPS.