  We consider the index of a Dirac operator on a compact even dimensional
manifold with a domain wall. The latter is defined as a co-dimension one
submanifold where the connection jumps. We formulate and prove an analog of the
Atiyah-Patodi-Singer theorem that relates the index to the bulk integral of
Pontryagin density and $\eta$-invariants of auxiliary Dirac operators on the
domain wall. Thus the index is expressed through the global chiral anomaly in
the volume and the parity anomaly on the wall.
