While the community of 3D point cloud generation has witnessed a big growth in recent years, there still lacks an effective way to enable intuitive user control in the generation process, hence limiting the general utility of such methods. Since an intuitive way of decomposing a shape is through its parts, we propose to tackle the task of controllable part-based point cloud generation. We introduce DiffFacto, a novel probabilistic generative model that learns the distribution of shapes with part-level control. We propose a factorization that models independent part style and part configuration distributions and presents a novel cross-diffusion network that enables us to generate coherent and plausible shapes under our proposed factorization. Experiments show that our method is able to generate novel shapes with multiple axes of control. It achieves state-of-the-art part-level generation quality and generates plausible and coherent shapes while enabling various downstream editing applications such as shape interpolation, mixing, and transformation editing.