We study the problem of generating point clouds of 3D objects. Instead of discretizing the object into 3D voxels with huge computational cost and resolution limitations, we propose a novel geometry image based generator (GIG) to convert the 3D point cloud generation problem to a 2D geometry image generation problem. Since the geometry image is a completely regular 2D array that contains the surface points of the 3D object, it leverages both the regularity of the 2D array and the geodesic neighborhood of the 3D surface. Thus, one significant benefit of our GIG is that it allows us to directly generate the 3D point clouds using efficient 2D image generation networks. Experiments on both rigid and non-rigid 3D object datasets have demonstrated the promising performance of our method to not only create plausible and novel 3D objects, but also learn a probabilistic latent space that well supports the shape editing like interpolation and arithmetic.