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 20122014 (and perhaps longer).
