  The action of any group on itself by conjugation and the corresponding
conjugacy relation play an important role in group theory. There have been
several attempts to extend the notion of conjugacy to semigroups. In this
paper, we present a new definition of conjugacy that can be applied to an
arbitrary semigroup and it does not reduce to the universal relation in
semigroups with a zero. We compare the new notion of conjugacy with existing
definitions, characterize the conjugacy in various semigroups of
transformations on a set, and count the number of conjugacy classes in these
semigroups when the set is infinite.
