  We consider the Adler-Bardeen anomaly of the U(1) axial current in abelian
and non-abelian gauge theories and present its algebraic characterization as
well as an explicit evaluation proving regularization scheme independence of
the anomaly. By extending the gauge coupling to an external space-time
dependent field we get a unique definition for the quantum corrections of the
topological term. It also implies a simple proof of the non-renormalization
theorem of the Adler-Bardeen anomaly. We consider local gauge couplings in
supersymmetric theories and find that there the renormalization of the gauge
coupling is determined by the topological term in all loop orders except for
one loop. It is shown that in one-loop order the quantum corrections to the
topological term induce an anomalous breaking of supersymmetry, which is
characterized by similar properties as the Adler-Bardeen anomaly.
