  We establish the Thom isomorphism in twisted K-theory for any real vector
bundle and develop the push-forward map in twisted K-theory for any
differentiable proper map $f: X\to Y$ (not necessarily K-oriented). The
push-forward map generalizes the push-forward map in ordinary K-theory for any
$K$-oriented differentiable proper map and the Atiyah-Singer index theorem of
Dirac operators on Clifford modules. For $D$-branes satisfying Freed-Witten's
anomaly cancellation condition in a manifold with a non-trivial $B$-field, we
associate a canonical element in the twisted K-group to get the so-called
D-brane charges.
