Deep generative models have shown success in generating 3D shapes with different representations. In this work, we propose Neural Volumetric Mesh Generator (NVMG), which can generate novel and high-quality volumetric meshes. Unlike the previous 3D generative model for point cloud, voxel, and implicit surface, volumetric mesh is a ready-to-use representation in industry with details on both the surface and interior. Generating this kind of highly-structured data thus brings a great challenge. To tackle this problem, we first propose to use a diffusion-based generative model to generate voxelized shapes with realistic shape and topology information. With the voxelized shape, we can simply obtain a tetrahedral mesh as a template. Further, we use a voxel-conditional neural network to predict the surface conditioned on the voxels, and progressively project the tetrahedral mesh to the predicted surface under regularization. As shown in the experiments, without any post-processing, our pipeline can generate high-quality artifact-free volumetric and surface meshes.