2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/127642This survey aims to cover the motivation for and history of the study of local rigidity of group actions. There is a particularly detailed discussion of recent results, including outlines of some proofs. The article ends with a large number of conjectures and open questions and aims to point to interesting directions for future research. {\em To Anatole Katok on the occasion of his 60th birthday.}Survey article, final version for publication, see also http://mypage.iu.edu/~fisherdm/Dynamical SystemsDifferential Geometry37C85; 53C24text