We show that various families of fishnet-type Feynman integrals are invariant under a Yangian over the conformal algebra. This observation comprises scalar triangle, square and hexagon graphs in three, four and six spacetime dimensions, respectively, as well as new types of integrable fishnets including fermions. The Yangian symmetry yields novel differential equations for these largely unsolved classes of integrals. Moreover, the considered fishnet graphs in three and four dimensions correspond to correlation functions and scattering amplitudes in specific double scaling limits of planar, gamma-twisted N=4 super Yang-Mills or ABJM theory. These limits define integrable quantum field theories in four and three spacetime dimensions and open the door to understanding the origins of AdS/CFT integrability.