  This report aims at giving a general overview on the classification of the
maximal subgroups of compact Lie groups (not necessarily connected). In the
first part, it is shown that these fall naturally into three types: (1) those
of trivial type, which are simply defined as inverse images of maximal
subgroups of the corresponding component group under the canonical projection
and whose classification constitutes a problem in finite group theory, (2)
those of normal type, whose connected one-component is a normal subgroup, and
(3) those of normalizer type, which are the normalizers of their own connected
one-component. It is also shown how to reduce the classification of maximal
subgroups of the last two types to: (2) the classification of the finite
maximal $\Sigma$-invariant subgroups of center-free connected compact simple
Lie groups and (3) the classification of the $\Sigma$-primitive subalgebras of
compact simple Lie algebras, where $\Sigma$ is a subgroup of the corresponding
outer automorphism group. In the second part, we explicitly compute the
normalizers of the primitive subalgebras of the compact classical Lie algebras
(in the corresponding classical groups), thus arriving at the complete
classification of all (non-discrete) maximal subgroups of the compact classical
Lie groups.
