  We present an overview and some new applications of the approximation by
conjugation method introduced by Anosov and the second author more than thirty
years ago \cite{AK}. Michel Herman made important contributions to the
development and applications of this method beginning from the construction of
minimal and uniquely ergodic diffeomorphisms jointly with Fathi in \cite{FH}
and continuing with exotic invariant sets of rational maps of the Riemann
sphere \cite{H3}, and the construction of invariant tori with nonstandard and
unexpected behavior in the context of KAM theory \cite{H1, H2}.
  Recently the method has been experiencing a revival. Some of the new results
presented in the paper illustrate variety of uses for tools available for a
long time, others exploit new methods, in particular possibility of mixing in
the context of Liouvillean dynamics discovered by the first author \cite{F1,
F2}.
