We discuss some new statistical methods for matching configurations of points in space where the points are either unlabelled or have at most a partial labelling constraining the match. The aim is to draw simultaneous inference about the matching and the transformation. Various questions arise: how to incorporate concommitant information? How to simulate realistic configurations? What are the implementation issues? What is the effect of multiple comparisons when a large data base is used?, and so on. Applications to protein bioinformatics, and image analysis will be described. We will also discuss some open problems and suggest directions for future work.