  We classify all real and strongly real classes of the finite special unitary
group $SU_n(q)$. Unless $q \equiv 3 (mod 4)$ and $n |4$, the classification of
real classes is similar to that of the finite special linear group $SL_n(q)$.
We relate strong reality in $SU_n(q)$ to strong reality in the finite special
orthogonal groups $SO^\pm_n(q)$, and we classify real and strongly real classes
in the last case for the group $SO^\pm_n(q)$, when q is odd and $n \equiv 2
(mod 4)$.
