In this work, we propose novel algorithms for inpainting and refinement of diffeomorphisms. The basic idea is to represent a diffeomorphism by its Beltrami coefficient. It is then possible to refine and inpaint the diffeomorphism by processing its Beltrami coefficient. With the inpainted/refined Beltrami coefficient, we can construct a new diffeomorphism using the exact Beltrami holomorphic flow algorithm proposed in this paper. We apply our algorithms on several practical applications, which include the inpainting of a highly distorted diffeomorphism, the inpainting of image sequences of deforming shapes, the super-resolution of diffeomorphisms and the global parameterization of cortical surfaces by combining local parameterizations. Experiments show that our algorithm can solve these problems with natural and smooth results. We demonstrates how our proposed method can be widely applied in areas from texture mapping to video processing, and from computer graphics to medical imaging.