  Twistor ideas have led to a number of recent advances in our understanding of
scattering amplitudes. Much of this work has been indirect, determining the
twistor space support of scattering amplitudes by examining the amplitudes in
momentum space. In this paper, we construct the actual twistor scattering
amplitudes themselves. We show that the recursion relations of Britto, Cachazo,
Feng and Witten have a natural twistor formulation that, together with the
three-point seed amplitudes, allows us to recursively construct general tree
amplitudes in twistor space. We obtain explicit formulae for $n$-particle MHV
and NMHV super-amplitudes, their CPT conjugates (whose representations are
distinct in our chiral framework), and the eight particle N^2MHV
super-amplitude. We also give simple closed form formulae for the N=8
supergravity recursion and the MHV and conjugate MHV amplitudes. This gives a
formulation of scattering amplitudes in maximally supersymmetric theories in
which superconformal symmetry and its breaking is manifest. For N^kMHV, the
amplitudes are given by 2n-4 integrals in the form of Hilbert transforms of a
product of $n-k-2$ purely geometric, superconformally invariant twistor delta
functions, dressed by certain sign operators. These sign operators subtly
violate conformal invariance, even for tree-level amplitudes in N=4 super
Yang-Mills, and we trace their origin to a topological property of split
signature space-time. We develop the twistor transform to relate our work to
the ambidextrous twistor diagram approach of Hodges and of Arkani-Hamed,
Cachazo, Cheung and Kaplan.
