Covering problems have a long history. In one such problem, we consider the set of discs of radius r, centered at the points of a unit intensity Poisson process in the plane, and ask for the probability that the union of these discs covers a fixed disc of area n>>r. In this talk, we will first address the basic question above, before discussing some of its extensions. Specifically, the *barrier coverage problem* concerns discs in a long horizontal strip of height h, and asks for the intensity of ``breaks" in coverage along the strip, while in the *secrecy coverage model*, the* *discs* *no longer have radius r, but instead depend on a second Poisson process; here we again ask for the probability that a large disc is completely covered. We have fairly accurate estimates in both cases, obtained by combining geometric ingredients with probabilistic tools such as the Chen-Stein method. An unusual feature is that, for both problems, the ``obvious" obstructions to coverage do not dominate. Finally, we will move on to some related percolation questions, where instead of precise results we have only bounds, conjectures and heuristics.

This is joint work with (various combinations of) Paul Balister, Bela Bollobas, Victor Falgas-Ravry, Martin Haenggi and Mark Walters.