  Ambitwistor strings are chiral, infinite tension analogues of conventional
string theory whose target space is the space of complex null geodesics and
whose spectrum consists exclusively of massless states. At genus zero, these
strings underpin the Cachazo-He-Yuan formulae for tree level scattering of
gravitons, gluons and scalars. In this paper we extend these formulae in a
number of directions. Firstly, we consider Ramond sector vertex operators and
construct simple amplitudes involving space-time fermions. These agree with
tree amplitudes in ten dimensional supergravity and super Yang--Mills. We then
show that, after the usual GSO projections, the ambitwistor string partition
function is modular invariant. We consider the scattering equations at genus
one, and calculate one loop scattering amplitudes for NS-NS external states in
the Type II ambitwistor string. We conjecture that these give new
representations of (the integrand of) one loop supergravity amplitudes and we
show that they have the expected behaviour under factorization of the
worldsheet in both non--separating and separating degenerations.
