  Ambitwistor strings are chiral (holomorphic) strings whose target is the
space of complex null geodesics, ambitwistor space. We introduce twistor
representations of ambitwistor space in 6 and 5 dimensions. In 6d the twistor
representation is naturally conformally invariant. Anomaly cancellation leads
to models that describe biadjoint scalar amplitudes and certain conformally
invariant gauge and gravity theories, respectively of 4$^{\rm th}$ and 6$^{\rm
th}$ order. There are three such models, reflecting triality for the conformal
group SO(8) associated to these 6d models. On reduction to five dimensions,
gauge anomaly cancellation requires supersymmetry and the resulting models
describe maximally supersymmetric Yang-Mills and gravity. The twistor
representation of these ambitwistor strings lead to formul{\ae} for maximally
supersymmetric gauge and gravity amplitudes based on the polarized scattering
equations in 5d, found earlier by the first two authors.
