Local Rigidity of group actions: Past, Present, Future

This 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.}