For certain natural sequences of topological spaces, the kth homology group stabilizes once you go far enough out in the sequence of spaces.  This phenomenon is called homological stability.  Two classical examples of homological stability are the configuration space of n unordered distinct points in the plane, studied in the 60's by Arnold' and the space of (based) algebraic maps from CP^1 to CP^1 studied by Segal in the 70's.  It turns out that the stable homology is the same in these two examples, and in this talk we explain that this is just the tip an iceberg--a subtle, but precise relationship between the values of stable of homology different sequences of spaces.  To explain this relationship, which we discovered through an analogy to asymptotic counts in number theory, we introduce a new notion of homological density.