Arnold's seminar October 12, 1999 I.Bogaevski "Mather's theory for Lagrange and Legendre mappings" In Mather's theory a smooth mapping is considered as a family of preimages of points of the image-manifold: two smooth mappings are RL-equivalent iff the corresponding families are V-equivalent as deformations. In particular, RL-stability of a smooth mapping is equivalent to V-versality of the corresponding family. Analogous facts are true for Lagrange (Legendre) mappings of Lagrange (Legendre) varieties. We will discuss a technique based on these facts and allowing to classify stable and generic germs of Lagrange and Legendre mappings. If there is enough time a full classification of stable simple Lagrange mappings of the open Whitney umbrella will be presented.