  We survey results related to the important Hossz\'u-Gluskin Theorem on
$n$-ary groups adding also several new results and comments. The aim of this
paper is to write all such results in uniform and compressive forms. Therefore
some proofs of new results are only sketched or omitted if their completing
seems to be not too difficult for readers. In particular, we show as the
Hossz\'u-Gluskin Theorem can be used for evaluation how many different $n$-ary
groups (up to isomorphism) exist on some small sets. Moreover, we sketch as the
mentioned theorem can be also used for investigation of
$\mathcal{Q}$-independent subsets of semiabelian $n$-ary groups for some
special families $\mathcal{Q}$ of mappings.
