  A geometric string solution has background fields in overlapping coordinate
patches related by diffeomorphisms and gauge transformations, while for a
non-geometric background this is generalised to allow transition functions
involving duality transformations. Non-geometric string backgrounds arise from
T-duals and mirrors of flux compactifications, from reductions with duality
twists and from asymmetric orbifolds. Strings in ` T-fold' backgrounds with a
local $n$-torus fibration and T-duality transition functions in $O(n,n;\Z)$ are
formulated in an enlarged space with a $T^{2n}$ fibration which is geometric,
with spacetime emerging locally from a choice of a $T^n$ submanifold of each
$T^{2n}$ fibre, so that it is a subspace or brane embedded in the enlarged
space. T-duality acts by changing to a different $T^n$ subspace of $T^{2n}$.
For a geometric background, the local choices of $T^n$ fit together to give a
spacetime which is a $T^n$ bundle, while for non-geometric string backgrounds
they do not fit together to form a manifold. In such cases spacetime geometry
only makes sense locally, and the global structure involves the doubled
geometry. For open strings, generalised D-branes wrap a $T^n$ subspace of each
$T^{2n}$ fibre and the physical D-brane is the part of the part of the physical
space lying in the generalised D-brane subspace.
