A well known classical theorem due to Bieberbach says
that every discrete group $\Gamma$ of isometries of the
$n$–dimensional Euclidean space $\mathbb R^n$ with compact
quotient $\Gamma\diagdown \mathbb R^n$ contains a subgroup of finite index consisting of translations. In particular such a group $\Gamma$ is virtually abelian, i.e. $\Gamma$ contains an abelian subgroup of finite index.\par L.Auslander conjectured that
every crystallographic subgroup $\Gamma$ of $G_n$ is virtually solvable, i.e. contains a solvable subgroup of finite index.\par
In our several joint (H.Abels, G.Margulis and G.Soifer) works, we made an approach to this problem. The main goal of this talk is to present ideas and methods introduced in this works as well as results which have been proved.