In this paper, we present a multigrid technique for efficiently deforming large surface and volume meshes. We show that a previous least-squares formulation for distortion minimization reduces to a Laplacian system on a general graph structure for which we derive an analytic expression. We then describe an efficient multigrid algorithm for solving the relevant equations. Here we develop novel prolongation and restriction operators used in the multigrid cycles. Combined with a simple but effective graph coarsening strategy, our multigrid algorithm outperforms other direct and multigrid solvers in both time and memory cost. While direct factorization methods have frequently been applied to related problems, it is demonstrated that even for modestly sized meshes, our multigrid solver surpasses the most sophisticated factorization codes. Moreover, since our multigrid solver does not rely on extensive precomputation, it is particularly well suited for integration with a general mesh editing environment where unpredictable combinations of different operations will invalidate the results of any such precomputation. Experimental evidence of these advantages is provided on a number of meshes with a wide range of size.