Voronoi conjecture for five-dimensional parallelohedra

In this talk I am going to discuss a well-known connection between lattices in $\mathbb{R}^d$ and convex polytopes that tile $\mathbb{R}^d$ 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 $\mathbb{R}^d$ 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.