I discuss a new family of four-dimensional CFTs, recently proposed by O.Gurdogan and myself, emerging as a double scaling limit of weakly coupled and strongly gamma-twisted N=4 SYM theory. These non-supersymmetric and non-unitary CFTs inherit the planar integrability of N=4 SYM and present a unique opportunity of non-perturbative study of four-dimensional conformal physics. Important physical quantities are dominated by a limited set of "chiral" Feynman graphs, such as "fishnet" graphs for the simplest, bi-scalar model. I present the results of exact calculation of some of these quantities, such as anomalous dimensions of local operators, some 3- and 4-point correlation functions and scattering amplitudes, by means of quantum spin chain techniques or the quantum spectral curve (QSC) approach originally proposed for N=4 SYM. The bi-scalar theory appears to be a remarkable "generator" of large families of integrable, i.e. computable multi-loop Feynman graphs.