  This paper is a survey, with few proofs, of ideas and notions related to
self-similarity of groups, semi-groups and their actions. It attempts to relate
these concepts to more familiar ones, such as fractals, self-similar sets, and
renormalizable dynamical systems. In particular, it presents a plausible
definition of what a "fractal group" should be, and gives many examples of such
groups.
  A particularly interesting class of examples, derived from monodromy groups
of iterated branch coverings, or equivalently from Galois groups of iterated
polynomials, is presented. This class contains interesting groups from an
algebraic point of view (just-non-solvable groups, groups of intermediate
growth, branch groups,...), and at the same time the geometry of the group is
apparent in that a limit of the group identifies naturally with the Julia set
of the covering map.
  In its survey, the paper discusses finite-state transducers, growth of groups
and languages, limit spaces of groups, hyperbolic spaces and groups, dynamical
systems, Hecke-type operators, C^*-algebras, random matrices, ergodic theorems
and entropy of non-commuting transformations. Self-similar groups appear then
as a natural weaving thread through these seemingly different topics.

Group Theory;Dynamical Systems