Voronoi conjecture for five-dimensional parallelohedra

In this talk I am going to discuss a well-known connection between lattices in and convex polytopes that tile with translations only.
My main topic will be the Voronoi conjecture, a century old conjecture which is, while stated in very simple terms, is still open in general. The conjecture states that every convex polytope that tiles with translations can be obtained as an affine image of the Voronoi domain for some lattice.
I plan to survey several known results on the Voronoi conjecture and give an insight on a recent proof of the Voronoi conjecture in the five-dimensional case. The talk is based on a joint work with Alexander Magazinov.