In this talk we review Voronoi's classical theory on perfect lattices, respectively on perfect (positive definite quadratic) forms, from the viewpoint of Ryshkov polyhedra. Based on it we explain how perfect lattices can (in principle) be classified in a given dimension $d$ and we provide some details on the recently finished classification for $d=8$. One of the main applications of such a classification is a solution of the lattice sphere packing problem. One may hope to find an answer to one of the major open problems on sphere packings: the existence of a dimension~$d$, for which there exist non-lattice packings which are denser than any lattice packing. For several dimensions $d\geq 10$ such packings are expected to exist, but so far the lattice sphere packing problem has only been solved for $d\leq 8$ and $d=24$ (where the Leech lattice likely gives the densest possible sphere packing). We introduce a way of extending Voronoi's theory to periodic sets which may help to shed some new light on the problem. On the one hand it yields computational tools for systematic future explorations of dense periodic sphere packings. On the other hand our framework allows to show that perfect and strongly eutactic lattices can not be locally improved to yield denser, periodic non-lattice packings.