In recent years, there has been significant process in the study of perturbative field theory scattering amplitudes using certain 2d chiral, first-order CFTs known as 'ambitwistor string theories.' After a brief review of the ambitwistor setup, I will introduce a related family of 2d CFTs, which can be viewed as constrained ambitwistor strings. These new models describe holomorphic maps from the Riemann sphere to the projective null cone in D-dimensional Minkowski space. This target space is the natural setting to describe field theories with (classical) conformal invariance in (D-2)-dimensions. Killing the anomalies associated with these models fixes critical dimensions for which three well-known space-time field theories (bi-adjoint cubic scalar, gauge theory, gravity) are conformal. Furthermore, the spectrum of each model contains all single field insertions, along with their conformal descendants, of the correct scaling dimension. Time permitting, I will also outline how the space-time 3-point functions can be obtained from the 3-point correlators on the Riemann sphere.