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Denoising diffusion models (DDMs) have shown promising results in 3D point cloud synthesis. To advance 3D DDMs and make them useful for digital artists, we require (i) high generation quality, (ii) flexibility for manipulation and applications such as conditional synthesis and shape interpolation, and (iii) the ability to output smooth surfaces or meshes. To this end, we introduce the hierarchical Latent Point Diffusion Model (LION) for 3D shape generation. LION is set up as a variational autoencoder (VAE) with a hierarchical latent space that combines a global shape latent representation with a point-structured latent space. For generation, we train two hierarchical DDMs in these latent spaces. The hierarchical VAE approach boosts performance compared to DDMs that operate on point clouds directly, while the point-structured latents are still ideally suited for DDM-based modeling. Experimentally, LION achieves state-of-the-art generation performance on multiple ShapeNet benchmarks. Furthermore, our VAE framework allows us to easily use LION for different relevant tasks: LION excels at multimodal shape denoising and voxel-conditioned synthesis, and it can be adapted for text- and image-driven 3D generation. We also demonstrate shape autoencoding and latent shape interpolation, and we augment LION with modern surface reconstruction techniques to generate smooth 3D meshes. We hope that LION provides a powerful tool for artists working with 3D shapes due to its high-quality generation, flexibility, and surface reconstruction. Project page and code: https://nv-tlabs.github.io/LION.