  This expository text is about using toric geometry and mirror symmetry for
evaluating Feynman integrals. We show that the maximal cut of a Feynman
integral is a GKZ hypergeometric series. We explain how this allows to
determine the minimal differential operator acting on the Feynman integrals. We
illustrate the method on sunset integrals in two dimensions at various loop
orders. The graph polynomials of the multi-loop sunset Feynman graphs lead to
reflexive polytopes containing the origin and the associated variety are
ambient spaces for Calabi-Yau hypersurfaces. Therefore the sunset family is a
natural home for mirror symmetry techniques. We review the evaluation of the
two-loop sunset integral as an elliptic dilogarithm and as a trilogarithm. The
equivalence between these two expressions is a consequence of 1) the local
mirror symmetry for the non-compact Calabi-Yau three-fold obtained as the
anti-canonical hypersurface of the del Pezzo surface of degree 6 defined by the
sunset graph polynomial and 2) that the sunset Feynman integral is expressed in
terms of the local Gromov-Witten prepotential of this del Pezzo surface.
