  Four-dimensional conformal fishnet theory is an integrable scalar theory
which arises as a double scaling limit of $\gamma$-deformed maximally
supersymmetric Yang-Mills. We give a perturbative reformulation of
$\gamma$-deformed super-Yang-Mills theory in twistor space, and implement the
double scaling limit to obtain a twistor description of conformal fishnet
theory. The conformal fishnet theory retains an abelian gauge symmetry on
twistor space which is absent in space-time, allowing us to obtain
cohomological formulae for scattering amplitudes that manifest conformal
invariance. We study various classes of scattering amplitudes in twistor space
with this formalism.
