  In this paper we introduce and develop the theory of FI-modules. We apply
this theory to obtain new theorems about:
  - the cohomology of the configuration space of n distinct ordered points on
an arbitrary (connected, oriented) manifold
  - the diagonal coinvariant algebra on r sets of n variables
  - the cohomology and tautological ring of the moduli space of n-pointed
curves
  - the space of polynomials on rank varieties of n x n matrices
  - the subalgebra of the cohomology of the genus n Torelli group generated by
H^1 and more.
  The symmetric group S_n acts on each of these vector spaces. In most cases
almost nothing is known about the characters of these representations, or even
their dimensions. We prove that in each fixed degree the character is given,
for n large enough, by a polynomial in the cycle-counting functions that is
independent of n. In particular, the dimension is eventually a polynomial in n.
In this framework, representation stability (in the sense of Church-Farb) for a
sequence of S_n-representations is converted to a finite generation property
for a single FI-module.
