  The purpose of this survey paper is to bring to a large mathematical audience
(containing also non-algebraists) some topics of invariant theory both in the
classical commutative and the recent noncommutative case. We have included only
several topics from the classical invariant theory -- the finite generating
(the Endlichkeitssatz) and the finite presenting (the Basissatz) of the algebra
of invariants, the Molien formula for its Hilbert series and the
Shephard-Todd-Chevalley theorem for the invariants of a finite group generated
by pseudo-reflections. Then we give analogues of these results for free and
relatively free associative and Lie algebras. Finally we deal with the algebra
of generic matrices and the invariant theory related with it.
