  Intuitively, the filter dimension of an algebra or a module measures how
`close' standard filtrations of the algebra or the module are. In particular,
for a simple algebra it also measures the growth of how `fast' one can prove
that the algebra is simple.
  The filter dimension appears naturally when one wants to generalize the
Bernstein's inequality for the Weyl algebras to the class of simple finitely
generated algebras.
  This paper is a review of the author's results on the filter dimension and
its connections with the Gelfand-Kirillov dimension, the Krull dimension, the
representation theory of simple finitely generated algebras and the `size' of
their maximal commutative subalgebras.
