  We show that tree level superstring theories on certain supersymmetric
backgrounds admit a symmetry which we call ``fermionic T-duality''. This is a
non-local redefinition of the fermionic worldsheet fields similar to the
redefinition we perform on bosonic variables when we do an ordinary T-duality.
This duality maps a supersymmetric background to another supersymmetric
background with different RR fields and a different dilaton. We show that a
certain combination of bosonic and fermionic T-dualities maps the full
superstring theory on $AdS_5 \times S^5$ back to itself in such a way that
gluon scattering amplitudes in the original theory map to something very close
to Wilson loops in the dual theory. This duality maps the ``dual superconformal
symmetry'' of the original theory to the ordinary superconformal symmetry of
the dual model. This explains the dual superconformal invariance of planar
scattering amplitudes of N=4 super Yang Mills and also sheds some light on the
connection between amplitudes and Wilson loops. In the appendix, we propose a
simple prescription for open superstring MHV tree amplitudes in a flat
background.
