  We consider the behaviour of fermions in the background of
instanton-anti\-instanton type configurations. Several different physics
problems, from the high energy electroweak interactions to the study of vacuum
structure of QCD and of large orders of perturbation theory are related to this
problem. The spectrum of the Dirac operator in such a background is studied in
detail. We present an approximation for the fermion correlation function when
the instanton-anti\-instanton separation ($R$) is large compared to their sizes
($\rho $). The situation when the instanton-anti\-instanton overlap and melt,
is studied through the behaviour of the Chern Simons number as a function of $
R/\rho$ and $x_4$.
  Applying our results to widely discussed cases of fermion-number violation in
the electroweak theory, we conclude that there are no theoretical basis for
expecting anomalous cross sections to become observable at energies in $10$ TeV
region.
