The notion of stability --speaking loosely, "sometimes an infinite sequence of vector spaces eventually starts being constant" -- appears in many branches of mathematics, perhaps most notably topology, where Harer's theorem about the stability of the homology of mapping class groups has driven decades of work. Some natural sequences of vector spaces are evidently NOT eventually constant: for instance, the space Q_n of quadratic polynomials in n variables has dimension (1/2)n(n-1), so gets larger and larger as n goes to infinity. On the other hand, Q_n carries an action of the symmetric group S_n by permutation of coordinates. We will discuss a new framework which allows us to speak meaningfully about what it means for a sequence of representations of S_n to be stable. It turns out that the structures we define are ubiquitous, appearing in topology (e.g. homology groups of configuration spaces and of moduli spaces of curves) algebraic combinatorics (e.g. the graded pieces of diagonal coinvariant algebras) and algebraic geometry (e.g. spaces of polynomials on discriminant and rank varieties.) We prove, for instance, that all these sequences of vector spaces have dimension which is eventually a polynomial in n.