Introduction: Multiple Dirichlet Series

This introductory article aims to provide a roadmap to many of the interrelated papers in this volume and to a portion of the field of multiple Dirichlet series, particularly emerging new ideas. It is both a survey of the recent literature, and an introduction to the combinatorial aspects of Weyl group multiple Dirichlet series, a class of multiple Dirichlet series that are not Euler products, but which may nevertheless be reconstructed from their p-parts. These p-parts are combinatorially interesting, and may often be identified with p-adic Whittaker functions.