  We study twisted $Spin^c$-manifolds over a paracompact Hausdorff space $X$
with a twisting $\alpha: X \to K(\ZZ, 3)$. We introduce the topological index
and the analytical index on the bordism group of $\alpha$-twisted
$Spin^c$-manifolds over $(X, \alpha)$, taking values in topological twisted
K-homology and analytical twisted K-homology respectively. The main result of
this paper is to establish the equality between the topological index and the
analytical index. We also define a notion of geometric twisted K-homology,
whose cycles are geometric cycles of $(X, \a)$ analogous to Baum-Douglas's
geometric cycles. As an application of our twisted index theorem, we discuss
the twisted longitudinal index theorem for a foliated manifold $(X, F)$ with a
twisting $\alpha: X \to K(\ZZ, 3)$, which generalizes the Connes-Skandalis
index theorem for foliations and the Atiyah-Singer families index theorem to
twisted cases.
