  The overlap formula for the chiral determinant is presented and the
realization of gauge anomalies and gauge field toplogy in this context is
discussed. The ability of the overlap formalism to deal with supersymmetric
theories and Majorana-Weyl fermions is outlined. Two applications of the
overlap formalism are discussed in some detail. One application is the
computation of a fermion number violating process in a two dimensional U(1)
chiral gauge theory. The second application is a measurement of the probability
ditribution of the index of the chiral Dirac operator in four dimensional pure
SU(2) lattice gauge theory.
