  Based on the construction of Poisson-Lie T-dual $\sigma$-models from a common
parent action we study a candidate for the non-abelian respectively Poisson-Lie
T-duality group. This group generalises the well-known abelian T-duality group
O(d,d) and we explore some of its subgroups, namely factorised dualities, B-
and $\beta$-shifts. The corresponding duality transformed $\sigma$-models are
constructed and interpreted as generalised (non-geometric) flux backgrounds.
  We also comment on generalisations of results and techniques known from
abelian T-duality. This includes the Lie algebra cohomology interpretation of
the corresponding non-geometric flux backgrounds, remarks on a double field
theory based on non-abelian T-duality and an application to the investigation
of Yang-Baxter deformations. This will show that homogeneously Yang-Baxter
deformed $\sigma$-models are exactly the non-abelian T-duality $\beta$-shifts
when applied to principal chiral models.
