The computation of Feynman integrals still represents a bottle neck for precision calculations in quantum field theory. Recent progress goes hand in hand with the understanding of new function classes and their mathematical structure. In this talk we discuss the so-called fishnet integrals that represent an infinite family of scalar Feynman integrals with integrable structures. In particular we review the Yangian quantum group symmetry of these integrals which extends the conformal spacetime symmetry of the associated fishnet quantum field theories. We then focus on the specific case of fishnets in two spacetime dimensions and argue that these integrals compute volumes of Calabi-Yau geometries. Here the Yangian provides a convenient tool that yields the Calabi-Yau differential operators, while the geometry dictates the particular linear combination of Yangian invariants which furnishes the Feynman integrals.