  A recent application of an index relation of the form, $dim\ ker\ M - dim\
ker\ M^{\dagger} = \nu$, to the generation of chiral fermions in a vector-like
gauge theory is reviewed. In this scheme the chiral structure arises from a
mass term with a non-trivial index.The essence of the generalized Pauli-Villars
regularization of chiral gauge theory, which is based on this mechanism, is
also clarified.
