  This is Chapter 24 in the "AutoMathA" handbook. Finite automata have been
used effectively in recent years to define infinite groups. The two main lines
of research have as their most representative objects the class of automatic
groups (including word-hyperbolic groups as a particular case) and automata
groups (singled out among the more general self-similar groups).
  The first approach implements in the language of automata some tight
constraints on the geometry of the group's Cayley graph, building strange,
beautiful bridges between far-off domains. Automata are used to define a normal
form for group elements, and to monitor the fundamental group operations.
  The second approach features groups acting in a finitely constrained manner
on a regular rooted tree. Automata define sequential permutations of the tree,
and represent the group elements themselves. The choice of particular classes
of automata has often provided groups with exotic behaviour which have
revolutioned our perception of infinite finitely generated groups.
