Voronin Universality in several complex variables

We prove the Voronin universality theorem for the multiple Hurwitz zeta-function with rational or transcendental parameters in $\mathbb{C}^n$ answering a question of Matsumoto. In particular this implies that the Euler-Zagier multiple zeta-function is universal in several complex variables and gives the first example of a Dirichlet series that is universal in more than one variable.