The millimeter-wave radar sensor maintains stable performance under adverse environmental conditions, making it a promising solution for all-weather perception tasks, such as outdoor mobile robotics. However, the radar point clouds are relatively sparse and contain massive ghost points, which greatly limits the development of mmWave radar technology. In this paper, we propose a novel point cloud super-resolution approach for 3D mmWave radar data, named Radar-diffusion. Our approach employs the diffusion model defined by mean-reverting stochastic differential equations(SDE). Using our proposed new objective function with supervision from corresponding LiDAR point clouds, our approach efficiently handles radar ghost points and enhances the sparse mmWave radar point clouds to dense LiDAR-like point clouds. We evaluate our approach on two different datasets, and the experimental results show that our method outperforms the state-of-the-art baseline methods in 3D radar super-resolution tasks. Furthermore, we demonstrate that our enhanced radar point cloud is capable of downstream radar point-based registration tasks.