  In this survey paper, we discuss the problem of characterizing the critical
sets of bounded analytic functions in the unit disk of the complex plane. This
problem is closely related to the Berger-Nirenberg problem in differential
geometry as well as to the problem of describing the zero sets of functions in
Bergman spaces. It turns out that for any non-constant bounded analytic
function in the unit disk there is always a (essentially) unique "maximal"
Blaschke product with the same critical points. These maximal Blaschke products
have remarkable properties simliar to those of Bergman space inner functions
and they provide a natural generalization of the class of finite Blaschke
products.
