I will discuss the properties of a 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-unitary CFTs inherit the integrability of N=4 SYM in the planar limit and present a unique opportunity of a non-perturbative study of four-dimensional conformal physics. Important physical quantities are dominated by a limited subset of 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 4point correlation functions and scattering amplitudes, by means of spin chain techniques or the quantum spectral curve (QSC) approach originally proposed for N=4 SYM.