  This is a survey of work in which the author was involved in recent years. We
consider C*-algebras constructed from representations of one or several
algebraic endomorphisms of a compact abelian group - or, dually, of a discrete
abelian group. In our survey we do not try to describe the entire scope of the
methods and results obtained in the original papers, but we concentrate on the
important thread coming from the action of the multiplicative semigroup of a
Dedekind ring on its additive group. Representations of such actions give rise
to particularly intriguing problems and the study of the corresponding
C*-algebras has motivated many of the new methods and general results obtained
in this area.
