  We introduce the idea of *representation stability* (and several variations)
for a sequence of representations V_n of groups G_n. A central application of
the new viewpoint we introduce here is the importation of representation theory
into the study of homological stability. This makes it possible to extend
classical theorems of homological stability to a much broader variety of
examples. Representation stability also provides a framework in which to find
and to predict patterns, from classical representation theory
(Littlewood--Richardson and Murnaghan rules, stability of Schur functors), to
cohomology of groups (pure braid, Torelli and congruence groups), to Lie
algebras and their homology, to the (equivariant) cohomology of flag and
Schubert varieties, to combinatorics (the (n+1)^(n-1) conjecture). The majority
of this paper is devoted to exposing this phenomenon through examples. In doing
this we obtain applications, theorems and conjectures.
  Beyond the discovery of new phenomena, the viewpoint of representation
stability can be useful in solving problems outside the theory. In addition to
the applications given in this paper, it is applied in [CEF] to counting
problems in number theory and finite group theory. Representation stability is
also used in [C] to give broad generalizations and new proofs of classical
homological stability theorems for configuration spaces on oriented manifolds.
