One of the well-known applications of crystallographic groups is  the classification
of flat complete Riemannian manifolds by their fundamental group, which is
a  *torsion-free* crystallographic group (aka Bieberbach group). A very nice book
about this is  "Spaces of constant Curvature" by Joseph A. Wolf.
There are many interesting generalizations in this direction. One is due to
John Milnor and Louis Auslander, so  called affine crystallographic groups. 
Here the Bieberbach theorems for crystallographic groups have been generalized, at least 
conjecturally. Every (Euclidean) crystallographic group is virtually abelan (the translations forming 
an abelian normal subgroup of finite index).
The generalization to affine crystallographic groups should be that such groups are virtually
polycyclic. In othe rwords, the fundamental group of a conplete compact affine manifold should
be virtually polycyclic. This is still an open conjecture, called *Auslander's conjecture*.
It has received a lot of attention, see the work of Abels, Margulis and Soifer, ranging from 1995 until 2014 (and perhaps longer).