Flavors of Translative Coverings
  We survey results on the problem of covering the space ${\mathbb R}^n$, or a convex body in it, by translates of a convex body. Our main goal is to present a diverse set of methods. A theorem of Rogers is a central result, according to which, for any convex body $K$, the space ${\mathbb R}^n$ can be covered by translates of $K$ with density around $n\ln n$. We outline four approaches to proving this result. Then, we discuss the illumination conjecture, decomposability of multiple coverings, Sudakov's inequality and some problems concerning coverings by sequences of sets. 
M\'arton Nasz\'odi
math.MG