In this paper, we propose a novel 3D registration paradigm, Generative Point Cloud Registration, which bridges advanced 2D generative models with 3D matching tasks to enhance registration performance. Our key idea is to generate cross-view consistent image pairs that are well-aligned with the source and target point clouds, enabling geometry-color feature fusion to facilitate robust matching. To ensure high-quality matching, the generated image pair should feature both 2D-3D geometric consistency and cross-view texture consistency. To this end, we introduce DepthMatch-ControlNet and LiDARMatch-ControlNet, two matching-specific, controllable 2D generative models. Specifically, for depth camera-based 3D registration with point clouds derived from the depth maps, DepthMatch-ControlNet leverages the depth-conditioned generation capabilities of ControlNet to synthesize perspective-view RGB images that are geometrically consistent with depth maps, ensuring accurate 2D-3D alignment. Additionally, by incorporating a coupled conditional denoising scheme and coupled prompt guidance, it further promotes cross-view feature interaction, guiding texture consistency generation. To address LiDAR-based 3D registration with point clouds captured by LiDAR sensors, LiDARMatch-ControlNet extends this framework by conditioning on paired equirectangular range maps projected from 360-degree LiDAR point clouds, generating corresponding panoramic RGB images. Our generative 3D registration paradigm is general and can be seamlessly integrated into a wide range of existing registration methods to improve their performance. Extensive experiments on the 3DMatch and ScanNet datasets (for depth-camera settings), as well as the Dur360BEV dataset (for LiDAR settings), demonstrate the effectiveness of our approach.