  We use supertwistor space to construct scattering amplitudes of maximal
superconformal theories in three and six dimensions. In both cases, the
constraints of superconformal invariance and rationality imply that the
three-point amplitude vanishes on-shell, which constrains the four-point
amplitude to have vanishing residues in all channels. In three dimensions, we
find a unique solution for the four-point amplitude and demonstrate that it
agrees with the component result in the BLG theory. This suggests that BLG is
the unique three-dimensional theory with classical OSp(8|4) symmetry that
admits a Lagrangian description. We also show that one can derive the
four-point amplitude of the ABJM theory from our N=8 result by reducing the
supersymmetry, which implies that the tree-level Yangian symmetry recently
found in ABJM is also present in BLG. In six dimensions, we find that the
consistency conditions imply that all tree-level amplitudes vanish. This leads
us to conjecture that an interacting six-dimensional theory with classical
OSp(8|4) symmetry does not have a Lagrangian description, local or nonlocal,
unless the (2,0) tensor multiplets are supplemented by additional degrees of
freedom.
