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Mesh is a fundamental representation of 3D assets in various industrial applications, and is widely supported by professional softwares. However, due to its irregular structure, mesh creation and manipulation is often time-consuming and labor-intensive. In this paper, we propose a highly controllable generative model, GetMesh, for mesh generation and manipulation across different categories. By taking a varying number of points as the latent representation, and re-organizing them as triplane representation, GetMesh generates meshes with rich and sharp details, outperforming both single-category and multi-category counterparts. Moreover, it also enables fine-grained control over the generation process that previous mesh generative models cannot achieve, where changing global/local mesh topologies, adding/removing mesh parts, and combining mesh parts across categories can be intuitively, efficiently, and robustly accomplished by adjusting the number, positions or features of latent points. Project page is https://getmesh.github.io.