  Let Phi be a reduced root system of rank r. A Weyl group multiple Dirichlet
series for Phi is a Dirichlet series in r complex variables s_1,...,s_r,
initially converging for Re(s_i) sufficiently large, that has meromorphic
continuation to C^r and satisfies functional equations under the
transformations of C^r corresponding to the Weyl group of Phi. A heuristic
definition of such series was given in [2], and they have been investigated in
certain special cases in [2-6, 11-14]. In this paper we generalize results in
[13] to construct Weyl group multiple Dirichlet series by a uniform method, and
show in all cases that they have the expected properties.
