  When a quantum field theory has a symmetry, global or local like in gauge
theories, in the tree or classical approximation formal manipulations lead to
believe that the symmetry can also be implemented in the full quantum theory,
provided one uses the proper quantization rules. While this is often true, it
is not a general property and therefore requires a proof because simple formal
manipulations ignore the unavoidable divergences of perturbation theory. The
existence of invariant regularizations allows solving the problem in most cases
but the combination of gauge symmetry and chiral fermions leads to subtle
issues. Depending on the specific group and field content, anomalies are found:
obstructions to the quantization of chiral gauge symmetries.Because anomalies
take the form of local polynomials in the fields, are linked to local group
transformations, but vanish for global (rigid) transformations they have a
topological character.In these notes we review various perturbative and
non-perturbative regularization techniques, and show that they leave room for
possible anomalies when both gauge fields and chiral fermions are present. We
determine the form of anomalies in simple examples. We relate anomalies to the
index of the Dirac operator in a gauge background. We exhibit gauge instantons
that contribute to the anomaly in the example of the CP(N-1) models and SU(2)
gauge theories. We briefly mentioned a few physical consequences. For many
years the problem of anomalies had been discussed only within the framework of
perturbation theory. New non-perturbative solutions based on lattice
regularization have recently been proposed. We describe the so-called overlap
and domain wall fermion formulations.
