  We construct analogues of FI-modules where the role of the symmetric group is
played by the general linear groups and the symplectic groups over finite rings
and prove basic structural properties such as Noetherianity. Applications
include a proof of the Lannes--Schwartz Artinian conjecture in the generic
representation theory of finite fields, very general homological stability
theorems with twisted coefficients for the general linear and symplectic groups
over finite rings, and representation-theoretic versions of homological
stability for congruence subgroups of the general linear group, the
automorphism group of a free group, the symplectic group, and the mapping class
group.
