  We study the geometry and Hodge theory of the cubic hypersurfaces attached to
two-loop Feynman integrals for generic physical parameters. We show that the
Hodge structure attached to planar two-loop Feynman graphs decomposes into
mixed Tate pieces and the Hodge structures of families of hyperelliptic,
elliptic, or rational curves depending on the space-time dimension. For
two-loop graphs with a small number of edges, we give more precise results. In
particular, we recover a result of Bloch arXiv:2105.06132 that in the
well-known double box example, there is an underlying family of elliptic
curves, and we give a concrete description of these elliptic curves.
  We argue that the motive for the non-planar two-loop tardigrade graph is that
of a K3 surface of Picard number 11 and determine the generic lattice
polarization. Lastly, we show that generic members of the ice cream cone family
of graph hypersurfaces correspond to pairs of sunset Calabi--Yau varieties.
