A long-standing open problem, known as Hadwiger’s covering problem, asks what is the smallest natural number N(n) such that every convex body in {\mathbb R}^n can be covered by a union of the interiors of at most N(n) of its translates.

In this talk, I will discuss some history of this problem and its close relatives, and present more recent results, including a new general upper bound for N(n).