  We prove several reality properties for finite simple orthogonal groups. For
any prime power $q$ and $m\geq 1$, we show that all real conjugacy classes are
strongly real in the simple groups $\mathrm{P}\Omega^{\pm}(4m+2,q), m \geq 1$,
except in the case $\mathrm{P}\Omega^{-}(4m+2,q)$ with $q \equiv 3(\mathrm{mod}
\; 4)$, and we construct weakly real classes in this exceptional case for any
$m$. We also show that no irreducible complex character of
$\mathrm{P}\Omega^{\pm}(n,q)$ can have Frobenius-Schur indicator $-1$, except
possibly in the case $\mathrm{P}\Omega^{-}(4m+2,q)$ with $q \equiv
3(\mathrm{mod} \; 4)$.
