  In this paper we apply the theory of finitely generated FI-modules developed
by Church, Ellenberg and Farb to certain sequences of rational cohomology
groups. Our main examples are the cohomology of the moduli space of n-pointed
curves, the cohomology of the pure mapping class group of surfaces and some
manifolds of higher dimension, and the cohomology of classifying spaces of some
diffeomorphism groups. We introduce the notion of FI[G]-module and use it to
strengthen and give new context to results on representation stability
discussed by the author in a previous paper. Moreover, we prove that the Betti
numbers of these spaces and groups are polynomial and find bounds on their
degree. Finally, we obtain rational homological stability of certain wreath
products.
