  We verify the self-duality of Green-Schwarz supercoset sigma models on AdS$_d
\times S^d $ backgrounds (d=2,3,5) under combined bosonic and fermionic
T-dualities without gauge fixing kappa symmetry. We also prove this property
for superstrings on AdS$_d \times S^d \times S^d$ (d=2,3) described by
supercoset sigma models with the isometries governed by the exceptional Lie
supergroups $D(2,1;\alpha)$ (d=2) and $D(2,1;\alpha)\times D(2,1;\alpha)$
(d=3), which requires an additional T-dualisation along one of the spheres.
Then, by taking into account the contribution of non-supercoset fermionic modes
(up to the second order), we provide evidence for the T-self-duality of the
complete type IIA and IIB Green-Schwarz superstring theory on AdS$_d\times S^d
\times T^{10-2d}$ (d=2,3) backgrounds with Ramond-Ramond fluxes. Finally,
applying the Buscher-like rules to T-dualising supergravity fields, we prove
the T-self-duality of the whole class of the AdS$_d\times S^d \times M^{10-2d}$
superbackgrounds with Ramond-Ramond fluxes in the context of supergravity.
