While the simplest Feynman diagrams evaluate to multiple polylogarithms, more complicated functions can arise, involving integrals over higher-dimensional manifolds. Surprisingly, all examples of such manifolds in the literature to date are Calabi-Yau. I discuss why this is, and prove that a specific class of "marginal" diagrams give rise to Calabi-Yau manifolds. I demonstrate a bound on the dimensionality of these manifolds with loop order, and present infinite families of diagrams that saturate this bound to all orders. Finally, I discuss some integrals that don't fit this picture and still need a good explanation.