We study the problem of covering Euclidean space by possibly overlapping translates of a convex body , such that almost every
point is covered exactly times, for a fixed integer . Such a covering of Euclidean space by translations is called a -tiling. We will first give a historical survey that includes the investigations of classical tilings by translations (which we call -tilings in this context). They began with the work of the famous crystallographer Fedorov and with the work of Minkowski, who founded the Geometry of Numbers. Some 50 years later Venkov and McMullen gave a complete characterization of all convex objects that -tile Euclidean space.
Today we know that -tilings can be tackled by methods from Fourier analysis, though some of their aspects can be studied using purely combinatorial means. For many of our results, there is both a combinatorial proof and a Harmonic analysis proof. For larger than , the collection of convex objects that -tile is much wider than the collection of objects that -tile, and there is currently no complete knowledge of the polytopes that -tile, even in dimensions. We will cover both ``ancient'', as well as very recent, results concerning -tilings and other -tilings. This is joint work with Nick Gravin and Dmitry Shiryaev.