Supervised 3D part segmentation models are tailored for a fixed set of objects and parts, limiting their transferability to open-set, real-world scenarios. Recent works haveexplored vision-language models (VLMs) as a promising alternative, using multi-view rendering and textual prompting to identify object parts. However, naively applying VLMs in this context introduces several drawbacks, such as the need for meticulous prompt engineering, and fails to leverage the 3D geometric structure of objects. To address these limitations, we propose COPS, a COmprehensive model for Parts Segmentation that blends the semantics extracted from visual concepts and 3D geometry to effectively identify object parts. COPS renders a point cloud from multiple viewpoints, extracts 2D features, projects them back to 3D, and uses a novel geometric-aware feature aggregation procedure to ensure spatial and semantic consistency. Finally, it clusters points into parts and labels them. We demonstrate that COPS is efficient, scalable, and achieves zero-shot state-ofthe-art performance across five datasets, covering synthetic and real-world data, texture-less and coloured objects, as well as rigid and non-rigid shapes. The code is available at https://3d-cops.github.io.