Many tensors that arise in Riemannian geometry are double forms: tensors with p+q indices that alternate in their first p indices and alternate in their last q indices. The Riemann curvature tensor, for example, is a 4-tensor that alternates in its first 2 indices and its last 2 indices. In this talk I will discuss the algebraic and differential properties of double forms, and I will explain how they can be used to study finite element approximations of curvature tensors. Along the way, I will highlight three key tools: a canonical decomposition of double forms, an integration-by-parts identity for double forms, and variational properties of curvature. This is joint work with Yakov Berchenko-Kogan and Michael Neunteufel.