Atoms are the building blocks of all material systems and the diversity in their spatial arrangement gives rise to interesting material phenomena. For example, crystal structures with cubic symmetries are often more ductile than those with a hexagonal lattice symmetry. On the other hand, the lack of long range order in the arrangement of atoms gives glasses their unique viscoelastic properties. The materials science community has developed numerous techniques to describe atomic structures at these two extremes, i.e. the infinitely periodic crystals and the completely disordered glasses. However, there do not exist any general quantitative techniques for the identification and the description of patterns formed by atoms along defects, such as dislocations, surfaces and interfaces. In these defective regions, the crystal symmetry is broken but the arrangement of atoms is not entirely disordered. The properties of defects, which control the macroscopic material phenomena, is directly related to the geometrical packing of atoms in these regions. In this talk, I will present the atomistic structure of grain boundaries (GBs) from a novel perspective - as a stacking of polyhedral structural units. I will also introduce novel geometric algorithms rooted in Voronoi computations and point pattern matching techniques. As GBs exhibit atomic structures ranging from ordered (e.g. (111) twin boundary) to highly disordered (high-angle general GBs) packings, the techniques described here are anticipated to provide a unifying geometrical framework for defects in crystalline as well as amorphous systems.