Multiple Dirichlet series (MDS) are Dirichlet series in several complex variables mainly used in studying various analytic properties within certain families of automorphic L-functions (e.g., the family of all quadratic twists of a fixed L-function). I will begin by discussing the main motivation for studying these objects, with particular focus on the MDS associated to moments of quadratic Dirichlet L-functions. We have a local-global compatibility, and when the MDS are associated to moments of quadratic L-series, their local components are determined by connecting moments of character sums of hyperelliptic curves of a given genus over finite fields to traces on spaces of automorphic forms - a comparison of Arthur-Selberg and Grothendieck-Lefschetz trace formulas. In particular, I will present a tight connection with the recent work of J. Bergstr\"om, C. Faber and G. van der Geer. (Joint work with V. Pasol)

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