  We argue that the scattering amplitudes in the maximally supersymmetric N=4
super-Yang-Mills theory possess a new symmetry which extends the previously
discovered dual conformal symmetry. To reveal this property we formulate the
scattering amplitudes as functions in the appropriate dual superspace.
Rewritten in this form, all tree-level MHV and next-to-MHV amplitudes exhibit
manifest dual superconformal symmetry. We propose a new, compact and Lorentz
covariant formula for the tree-level NMHV amplitudes for arbitrary numbers and
types of external particles. The dual conformal symmetry is broken at loop
level by infrared divergences. However, we provide evidence that the anomalous
contribution to the MHV and NMHV superamplitudes is the same and, therefore,
their ratio is a dual conformal invariant function. We identify this function
by an explicit calculation of the six-particle amplitudes at one loop. We
conjecture that these properties hold for all, MHV and non-MHV, superamplitudes
in N=4 SYM both at weak and at strong coupling.
