  Let $S$ be a semigroup. The elements $a,b\in S$ are called primarily
conjugate if $a=xy$ and $b=yx$ for certain $x,y\in S$. The relation of
conjugacy is defined as the transitive closure of the relation of primary
conjugacy. In the case when $S$ is a monoid, denote by $G$ the group of units
of $S$. Then the relation of $G$-conjugacy is defined by $a\sim_G b \iff
a=g^{-1}bg$ for certain $g\in G$. We establish the structure of conjugacy
classes for regular epigroups (i.e. semigroups such that some power of each
element lies in a subgroup). As a corollary we obtain a criterion of conjugacy
in terms of $G$-conjugacy for factorizable inverse epigroups. We show that our
general conjugacy criteria easily lead to known and new conjugacy criteria for
some specific semigroups, among which are the full transformation semigroup and
the full inverse symmetric semigroup over a finite set, the linear analogues of
these semigroups and the semigroup of finitary partial automatic
transformations over a finite alphabet.
