It is well-known that a nilpotent n xn matrix B is determined up to conjugacy by a partition of n formed by the sizes of the Jordan blocks in the Jordan canonical form of B. We call this partition the Jordan type of B In the recent years several authors have worked on the following problem: for any partition P of n describe the type Q(P) of a generic nilpotent matrix commuting with a given nilpotent matrix of type P. In this talk we give an overview of the results on this problem obtained by studying the combinatorics of a poset associated to the partition P.