  We review double field theory (DFT) with emphasis on the doubled spacetime
and its generalized coordinate transformations, which unify diffeomorphisms and
b-field gauge transformations. We illustrate how the composition of generalized
coordinate transformations fails to associate. Moreover, in dimensional
reduction, the O(d,d) T-duality transformations of fields can be obtained as
generalized diffeomorphisms. Restricted to a half-dimensional subspace, DFT
includes `generalized geometry', but is more general in that local patches of
the doubled space may be glued together with generalized coordinate
transformations. Indeed, we show that for certain T-fold backgrounds with
non-geometric fluxes, there are generalized coordinate transformations that
induce, as gauge symmetries of DFT, the requisite O(d,d;Z) monodromy
transformations. Finally we review recent results on the \alpha' extension of
DFT which, reduced to the half-dimensional subspace, yields intriguing
modifications of the basic structures of generalized geometry.
