The moments of the uniform distribution on a convex polytope are rational functions in its vertex coordinates. We study the projective varieties parametrized by these moments.

This is work with Kathlen Kohn and Boris Shapiro. On our journey, we encounter Hankel determinantal ideals, splines, cumulants, multisymmetric functions, and invariants of nonreductive groups. While moment varieties are complicated, they offer many nice open problems.