By a classical result of Brodmann it is known that in any Noetherian ring, the set of associated prime ideals Ass(I^s) for the powers of an ideal I stabilizes for s>>0. In other words, there exists an integer s_0 such that Ass(I^s) = Ass(I^(s+1)) for all s >= s_0. This stable set of associated prime ideals is denoted by Ass^infty(I). The smallest such integer s_0 is called the index of stability. In this lecture we discuss the following questions: (i) Which finite sets of monomial prime ideals are of the form Ass^infty(I) for a suitable (squarefree) monomial ideal I? (ii) Is there a global bound of the index of stability? It can be shown that for any finite set P of nonzero monomial prime ideals there exists a monomial ideal I such that P = Ass^infty(I). However, an answer to (i) in the squarefree case is widely open. We give explicit descriptions of Ass^infty(I) for certain cases of matroidal and polymatroidal ideals. There is no example known of a monomial ideal in the polynomial ring in n variables whose index of stability is >= n. Thus we expect that this index is always