I will explain how free resolutions of ideals can be used to systematically formulate invariant theory for several moduli spaces of varieties that are of interest in arithmetic statistics and computational number theory. In particular, we extend the classical invariant theory formulas for the Jacobian of a genus one curve of degree n=2,3,4,5 to curves of arbitrary degree, generalizing the work on genus one models of Cremona, Fisher and Stoll, and in a joint work with Tom Fisher, we compute structure constants for a rank n ring from the free resolution of its associated set of n points in projective space, generalizing the previously known constructions of Levi-Delone-Faddeev and Bhargava. Time permitting I will talk about an ongoing project to extend these results to abelian varieties of higher dimension.