  In this paper we describe conjugacy classes of a Renner monoid $R$ with unit
group $W$, the Weyl group. We show that every element in $R$ is conjugate to an
element $ue$ where $u\in W$ and $e$ is an idempotent in a cross section
lattice. Denote by $W(e)$ and $W_*(e)$ the centralizer and stabilizer of $e\in
\Lambda$ in $W$, respectively. Let $W(e)$ act by conjugation on the set of left
cosets of $W_*(e)$ in $W$. We find that $ue$ and $ve$ ($u, v\in W$) are
conjugate if and only if $uW_*(e)$ and $vW_*(e)$ are in the same orbit. As
consequences, there is a one-to-one correspondence between the conjugacy
classes of $R$ and the orbits of this action. We then obtain a formula for
calculating the number of conjugacy classes of $R$, and describe in detail the
conjugacy classes of the Renner monoid of some $\cal J$-irreducible monoids.
  We then generalize the Munn conjugacy on a rook monoid to any Renner monoid
and show that the Munn conjugacy coincides with the semigroup conjugacy, action
conjugacy, and character conjugacy. We also show that the number of
inequivalent irreducible representations of $R$ over an algebraically closed
field of characteristic zero equals the number of the Munn conjugacy classes in
$R$.
