This is a survey on two important problems for isometries. The first one concerns the conservative distances, i.e., description of mappings between metric (normed, euclidean) spaces preserving (in one or in both directions) a fixed distance. Mainly the results of Alexandrov, Beckmann and Quarles, Ciesielski, Mielnik, Šemrl and Rassias are presented.
The second part concerns the stability of functional equations and the stability of isometries in particular. It brings a review of results of Hyers and Ulam, Bourgin, Gruber, Gevirtz, Dolinar, Rassias and others. A particular emphasis is put on contributions to the above problems by Th. M. Rassias to whom the paper is dedicated.