Multiple Dirichlet series are generalizations of L-functions
involving several complex variables. While the functional equation
of a usual L-series is an involution s -> 1-s, a multiple
Dirichlet series satisfies a group of functional equations that
intermixes all the variables.
In this talk we give examples of multiple Dirichlet series and their
applications. We will show how consideration of these series arises
naturally in number theoretic applications. Then we describe a
construction of Weyl group multiple Dirichlet series. These are
series attached to root systems plus some extra data where the
resulting group of functional equations is the associated Weyl group.
In general these series are expected to be Whitaker-Fourier
coefficients of metaplectic Eisenstein series, although that is not
how the series are defined. An essential part of the construction is
a deformation of the Weyl character formula for the associated
semisimple complex Lie algebra.