  Using a recently developed formulation of double field theory in superspace,
the graviton, $B$-field, gravitini, dilatini, and Ramond-Ramond bispinor are
encoded in a single generalized supervielbein. Duality transformations are
encoded as orthosymplectic transformations, extending the bosonic $O(D,D)$
duality group, and these act on all constituents of the supervielbein in an
easily computable way. We first review conventional non-abelian T-duality in
the Green-Schwarz superstring and describe the dual geometries in the language
of double superspace. Since dualities are related to super-Killing vectors,
this includes as special cases both abelian and non-abelian fermionic
T-duality.
  We then extend this approach to include Poisson-Lie T-duality and its
generalizations, including the generalized coset construction recently
discussed in arXiv:1912.11036. As an application, we construct the
supergeometries associated with the integrable $\lambda$ and $\eta$
deformations of the $AdS_5 \times S^5$ superstring. The deformation parameters
$\lambda$ and $\eta$ are identified with the possible one-parameter embeddings
of the supergravity frame within the doubled supergeometry. In this framework,
the Ramond-Ramond bispinors are directly computable purely from the algebraic
data of the supergroup.
