Indeterminate moment problems and the theory of entire functions

The set of solutions to an indeterminate Hamburger moment problem is given by the Nevanlinna parametrization involving four entire functions A,B,C,D. We review the properties of these functions, and discuss recent progress in the still open problem of characterizing the class of entire functions which occur in this way. They belong to the Krein class and are of minimal exponential type. uce the analogous parametrization of the set of solutions to an indeterminate Stieltjes moment problem from the corresponding symmetric Hamburger problem. The survey ends with a discussion of three special cases, where the functions A,B,C,D are known explicitly.