Dirichlet series are analytic tools similar to generating functions that number theorists use to study questions in arithmetic. The most famous example is the Riemann zeta function, a complex-valued function that encodes information about the distribution of prime numbers. In this talk, we review some of the nice properties of the Riemann zeta function before describing a class of multivariable Dirichlet series. The coefficients of these Weyl group multiple Dirichlet series both encode number theoretic data and satisfy combinatorial recurrence relations. We explain these features and some applications.