  We investigate fixed points and cycle types of permutation polynomials and
complete permutation polynomials arising from reversed Dickson polynomials of
the first kind and second kind over $\mathbb{F}_p$. We also study the
permutation behaviour of reversed Dickson polynomials of the first kind and
second kind over $\mathbb{Z}_m$. Moreover, we prove two special cases of a
conjecture on the permutation behaviour of reversed Dickson polynomials over
$\mathbb{F}_p$.
