  I review results from recent investigations of anomalies in fermion--Yang
Mills systems in which basic notions from noncommutative geometry (NCG) where
found to naturally appear. The general theme is that derivations of anomalies
from quantum field theory lead to objects which have a natural interpretation
as generalization of de Rham forms to NCG, and that this allows a geometric
interpretation of anomaly derivations which is useful e.g. for making these
calculations efficient. This paper is intended as selfcontained introduction to
this line of ideas, including a review of some basic facts about anomalies. I
first explain the notions from NCG needed and then discuss several different
anomaly calculations: Schwinger terms in 1+1 and 3+1 dimensional current
algebras, Chern--Simons terms from effective fermion actions in arbitrary odd
dimensions. I also discuss the descent equations which summarize much of the
geometric structure of anomalies, and I describe that these have a natural
generalization to NCG which summarize the corresponding structures on the level
of quantum field theory.
  Contribution to Proceedings of workshop `New Ideas in the Theory of
Fundamental Interactions', Szczyrk, Poland 1995; to appear in Acta Physica
Polonica B.
