  We classify all strongly real conjugacy classes of the finite unitary group
$\U(n, F_q)$ when $q$ is odd. In particular, we show that $g \in \U(n, F_q)$ is
strongly real if and only if $g$ is an element of some embedded orthogonal
group $O^{\pm}(n, F_q)$. Equivalently, $g$ is strongly real in $\U(n, F_q)$ if
and only if $g$ is real and every elementary divisor of $g$ of the form $(t \pm
1)^{2m}$ has even multiplicity. We apply this to obtain partial results on
strongly real classes in the finite symplectic group $\Sp(2n, F_q)$, $q$ odd,
and a generating function for the number of strongly real classes in $\U(n,
F_q)$, $q$ odd, and we also give partial results on strongly real classes in
$\U(n, F_q)$ when $q$ is even.
