(Joint work with Y. Barnea, M. Ershov, A. Le Boudec, M. Vannacci and Th. Weigel.) The commensurator of a group encapsulates (up to a suitable equivalence) all isomorphisms between finite index subgroups of the group. We study the commensurator of a free group F and of a free pro-p group, and also the p-commensurator of F (which is the subgroup of the commensurator that respects the pro-p topology on F), with a focus on normal subgroup structure. As well as 'global' results about the commensurator as a whole, we obtain some new constructions of simple groups: finitely generated simple groups with a free commensurated subgroup, and nondiscrete compactly generated simple locally compact groups that possibly have a free pro-p open subgroup.