There are many situations in topology where the homology type of a space is fixed or is dependent only on coarse combinatorial data whereas the fundamental group is a rich source of complexity. For example, in knot theory or in the study of the algebraic plane curves complements. Furthermore, in studying deformations of such embeddings, typically the homology groups of the exteriors do not vary, or are controlled by the combinatorics of the allowable singularities, whereas the fundamental group varies with few obvious constraints. Therefore to define interesting topological invariants of such embeddings, or of certain deformation classes of embeddings, it is vital to understand to what extent the homology of a space constrains its fundamental group. These issues are often profitably studied in purely group-theoretic terms. What aspects of a group are unchanged, or stable, under homology equivalences? The model theorem in this regard is the landmark 1963 result of J. Stallings (below) that each quotient, A/A_n, of a group by any term of its lower central series is preserved under any homological equivalence. Bill Dwyer later improved on Stallings' results. More recently, Shelly Harvey invented a superseries of the derived series and, together with me, proved analaogues of the Stallings and Dwyer theorems. In this talk we introduce a new notion, that of the stability of a series of groups under a class of maps, which gives a common framework in which all of these theorems can be understood. The work is joint with Shelly Harvey.