  The Index theorem for holomorphic line bundles on complex tori asserts that
some cohomology groups of a line bundle vanish according to the signature of
the associated hermitian form. In this article, this theorem is generalized to
quasi-tori, i.e. connected complex abelian Lie groups which are not necessarily
compact. In view of the Remmert-Morimoto decomposition of quasi-tori as well as
the K\"unneth formula, it suffices to consider only Cousin-quasi-tori, i.e.
quasi-tori which have no non-constant holomorphic functions. The Index theorem
is generalized to holomorphic line bundles, both linearizable and
non-linearizable, on Cousin-quasi-tori using $L^2$-methods coupled with the
Kazama-Dolbeault isomorphism and Bochner-Kodaira formulas.
