Place N points uniformly at random in a square of area N, where N is a large number. Now imagine growing a disc around each point, at the same speed for every point. Initially, there will be isolated discs, and small clusters of discs, surrounded by a sea of empty space, but then, as the disc radius r reaches about 0.6, a mysterious phenomenon known aspercolation occurs. Much later, as r reaches the coverage threshold, the discs entirely cover the original square. Both the percolation and coverage thresholds, for this model and others, are of interest to physicists, chemists, biologists and engineers, as well as mathematicians; come to the talk to find out why.