My happy ending problem

A set system is a k-fold covering of space if every point is contained in at least k sets. A 1-fold covering is called simply a covering. In 1980, motivated by a question of Laszlo Fejes Toth, I raised the following question. Given a plane convex set C, does there exist an integer k=k(C) such that every k-fold covering of the plane splits into 2 coverings? The same question makes sense in higher dimension. This problem has turned out to be relevant in sensor network scheduling and has generated a lot of research during the past 3 and a half decades. In this talk, I will summarize the curious history of the problem, and give a survey of the most important results and related open problems.