  In this paper we develop machinery for studying sequences of representations
of any of the three families of classical Weyl groups, extending work of
Church, Ellenberg, Farb, and Nagpal on the symmetric groups S_n to the signed
permutation groups B_n and the even-signed permutation groups D_n. For each
family W_n, we present an algebraic framework where a sequence V_n of
W_n-representations is encoded into a single object we call an FI_W-module. We
prove that if an FI_W-module V satisfies a simple finite generation condition
then the structure of the sequence is highly constrained. One consequence is
that the sequence is uniformly representation stable in the sense of
Church-Farb, that is, the pattern of irreducible representations in the
decomposition of each V_n eventually stabilizes in a precise sense. Using the
theory developed here we obtain new results about the cohomology of generalized
flag varieties associated to the classical Weyl groups, and more generally the
r-diagonal coinvariant algebras.
  We analyze the algebraic structure of the category of FI_W-modules, and
introduce restriction and induction operations that enable us to study
interactions between the three families of groups. We use this theory to prove
analogues of Murnaghan's 1938 stability theorem for Kronecker coefficients for
the families B_n and D_n. The theory of FI_W-modules gives a conceptual
framework for stability results such as these.
