For each nonnegative integer N, Miranda constructed a coarse moduli space of elliptic surfaces with section over the projective line with fundamental invariant N. I will explain how to compute the Chow rings with rational coefficients of these moduli spaces when N is at least 2. The Chow rings exhibit many properties analogous to those expected for the tautological ring of the moduli space of curves: they satisfy analogues of Faber’s conjectures, and they exhibit a stability property as N goes to infinity. When N=2, these elliptic surfaces are K3 surfaces polarized by a hyperbolic lattice. I will explain how the computation of the Chow ring confirms a special case of a conjecture of Oprea and Pandharipande on the structure of the tautological rings of moduli spaces of lattice polarized K3 surfaces. This is joint work with Bochao Kong.