  This article extends classical one variable results about Euler products
defined by integral valued polynomial or analytic functions to several
variables. We show there exists a meromorphic continuation up to a presumed
natural boundary, and also give a criterion, a la Estermann-Dahlquist, for the
existence of a meromorphic extension to $\C^n.$ Among applications we deduce
analytic properties of height zeta functions for toric varieties over $\Q$ and
group zeta functions.
