Abstract: Let $\Phi$ be a reduced root system of rank $r$. A {\it Weyl group multiple Dirichlet series} for $\Phi$ is a Dirichlet series in $r$ complex variables $s_1,\dots,s_r$, initially converging for $\re(s_i)$ sufficiently large, that has meromorphic continuation to ${\mathbb C}^r$ and satisfies functional equations under the transformations of ${\mathbb C}^r$ corresponding to the Weyl group of $\Phi$. A heuristic definition of such series was given in \cite{wmd1}, and they have been investigated in certain special cases in \cite{wmd1, wmd2, wmd3, wmd4, wmd5, qmds, cfg, nmdsA2, chbq}. In this paper we generalize results in \cite{nmdsA2} to construct Weyl group multiple Dirichlet series by a uniform method, and show in all cases that they have the expected properties.