Let R = K[x1, . . . , xn] be a polynomial ring in n variables over a field K and I be a monomial ideal of R. Let astab(I) and dstab(I) be the smallest integers l and k, for which Ass(I l ) and depth(R/I k ) stabilize, respectively. In this presentation, the goal is to introduce and study some basic concepts from combinatorial commutative algebra. In particular, we concentrate on property of polymatroidal ideals and astab(I) and dstab(I).