This talk will begin with a review of some standard notions of stabilization of 4-manifolds, two celebrated results of Wall from the 1960s, and Freedman's topological classification of closed, oriented simply-connected 4-manifolds from the 1980s.  We will then introduce what it means for a topological 4-manifold to be fertile – meaning (roughly) it has loads of smooth structures – and for it to be homologically fertile – meaning all its (primitive, ordinary) 2-dimensional homology classes can be represented (in some smooth structure) by loads of smoothly embedded spheres.  Finally, we'll indicate how gauge theory and symplectic topology show that many smoothable 4-manifolds are fertile, and explain how it follows that their stabilizations are homologically fertile.