I will survey algorithms for testing whether two point sets are congruent, that is, equal up to a Euclidean isometry. I will introduce the important techniques for congruence testing, namely dimension reduction and pruning, or more generally, condensation. I will illustrate these techniques on the three-dimensional version of the problem. I will then discuss algorithms for higher dimensions, and in particular, highlight the unpublished contributions of Jiří Matoušek. Finally, I will sketch our recent algorithm for four dimensions with near-linear running time (joint work with Heuna Kim).