  We develop ambitwistor string theories for 4 dimensions to obtain new
formulae for tree-level gauge and gravity amplitudes with arbitrary amounts of
supersymmetry. Ambitwistor space is the space of complex null geodesics in
complexified Minkowski space, and in contrast to earlier ambitwistor strings,
we use twistors rather than vectors to represent this space. Although
superficially similar to the original twistor string theories of Witten,
Berkovits and Skinner, these theories differ in the assignment of worldsheet
spins of the fields, rely on both twistor and dual twistor representatives for
the vertex operators, and use the ambitwistor procedure for calculating
correlation functions. Our models are much more flexible, no longer requiring
maximal supersymmetry, and the resulting formulae for amplitudes are simpler,
having substantially reduced moduli. These are supported on the solutions to
the scattering equations refined according to MHV degree and can be checked by
comparison with corresponding formulae of Witten and of Cachazo and Skinner.
