  Global properties of abelian noncommutative gauge theories based on
$\star$-products which are deformation quantizations of arbitrary Poisson
structures are studied. The consistency condition for finite noncommutative
gauge transformations and its explicit solution in the abelian case are given.
It is shown that the local existence of invertible covariantizing maps (which
are closely related to the Seiberg-Witten map) leads naturally to the notion of
a noncommutative line bundle with noncommutative transition functions. We
introduce the space of sections of such a line bundle and explicitly show that
it is a projective module. The local covariantizing maps define a new star
product $\star'$ which is shown to be Morita equivalent to $\star$.
