© 2006 Canadian Conference on Computational Geometry. All rights reserved. Given two sets A and B of n points each in R2, we study the problem of computing a matching between A and B that minimizes the root mean square (rms) distance of matched pairs. We can compute an optimal matching in O(n2+δ) time, for any δ > 0, and an ε-approximation in time O((n/ε)3/2 log6 n). If the set B is allowed to move rigidly to minimize the rms distance, we can compute a rigid motion of B and a matching in O((n4/ε5/2) log6 n) time whose cost is within (1 + ε) factor of the optimal one.