  Since the work of Ornstein and Weiss in 1987 (J. Analyse Math. 48 (1987)) it
has been understood that the natural category for classical ergodic theory
would be probability measure preserving actions of discrete amenable groups. A
conclusion of this work is that all such actions on nonatomic Lebesgue
probability spaces were orbit equivalent. From this foundation two broad
developements have been built. First, a full generalization of the various
equivalence theories, including Ornstein's isomorphism theorem itself, exists.
Second, one can use the characterization of discrete amenable actions as those
which are orbit equivalent to a action of \Z to lift theorems from actions of
\Z to those of arbitrary amenable groups.
