Multiple Dirichlet series and automorphic forms

This article gives an introduction to the multiple Dirichlet series arising from sums of twisted automorphic L-functions. We begin by explaining how such series arise from Rankin-Selberg constructions. Then more recent work, using Hartogs' continuation principle as extended by Bochner in place of such constructions, is described. Applications to the nonvanishing of L-functions and to other problems are also discussed, and a multiple Dirichlet series over a function field is computed in detail.