  We revisit the known problem whether each compact topology is contained in a
maximal compact topology and collect some partial answers to this question. For
instance we show that each compact topology is contained in a compact topology
in which convergent sequences have unique limits. We also answer a question of
D.E. Cameron by showing that each sequentially compact topology is contained in
a maximal sequentially compact topology. We finally observe that each sober
compact T_1-topology is contained in a maximal compact topology and that each
sober compact T_1-topology which is locally compact or sequential is the
infimum of a family of maximal compact topologies.
