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Visual grounding on 3D point clouds is an emerging vision and language task that benefits various applications in understanding the 3D visual world. By formulating this task as a grounding-by-detection problem, lots of recent works focus on how to exploit more powerful detectors and comprehensive language features, but (1) how to model complex relations for generating context-aware object proposals and (2) how to leverage proposal relations to distinguish the true target object from similar proposals are not fully studied yet. Inspired by the well-known transformer architecture, we propose a relation-aware visual grounding method on 3D point clouds, named as 3DVG-Transformer, to fully utilize the contextual clues for relationenhanced proposal generation and cross-modal proposal disambiguation, which are enabled by a newly designed coordinate-guided contextual aggregation (CCA) module in the object proposal generation stage, and a multiplex attention (MA) module in the cross-modal feature fusion stage. We validate that our 3DVG-Transformer outperforms the state-of-the-art methods by a large margin, on two point cloud-based visual grounding datasets, ScanRefer and Nr3D/Sr3D from ReferIt3D, especially for complex scenarios containing multiple objects of the same category.