  In this report, I discuss the current state of the problem of the axial
anomaly in quantum electrodynamics (QED) and quantum chromodynamics (QCD) and
how the axial anomaly is related to the structure of the vacuum in QCD. In QCD,
the vacuum average of the axial anomaly is proportional to a new quantum number
n, the winding number. The axial anomaly condition implies that there are zero
modes of the Dirac equation for a massless quark and that there is spontaneous
breaking of chiral symmetry in QCD, which leads to the formation of a quark
condensate. The axial anomaly can be represented in the form of a sum rule the
structure function in the dispersion representation of the axial -- vector --
vector (AVV) vertex. On the basis of this sum rule, it is calculated the width
of the \pi^0\to 2\gamma decay with an accuracy of 1.5%. It is demonstrated,
that 't Hooft conjecture -- the singularities of the amplitudes calculated in
perturbative QCD on quark-gluon basis should reproduce themselves in
calculations on the hadrons basis -- is not fulfilled generally.
