  We determine for which known finite simple groups $G$ and which primes $p$
the $p$-fusion system of $G$ is simple. This means first collecting together
the results that were already known (and correcting two errors made in an
earlier study of this question), and then handling the remaining cases. At the
same time, we develop some new tools to use when determining
$O^{p'}(\mathcal{F})$ for arbitrary saturated fusion systems.
