I will survey a class of problems with the following general theme:
Given a set of N overlapping geometric objects, such as disks or
squares, in the plane, consider their union. A vertex on the
boundary of the union is a point where boundaries of two objects
meet. The complexity of the union is the number of vertices on its
boundary. Let $c(N)$ be maximum such complexity, over all possible
choices of N objects. We are interested in asymptotic behavior of
$c(N)$ as a function of N, for a variety of object classes, in two and
higher dimensions.

Investigation of these functions has connections to geometric
algorithms on the one hand and to a variety of geometric,
combinatorial, and topological questions on the other.