  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.
