  String backgrounds with a local torus fibration such as T-folds are naturally
formulated in a doubled formalism in which the torus fibres are doubled to
include dual coordinates conjugate to winding number. Here we formulate and
explore a generalisation of this construction in which all coordinates are
doubled, so that the doubled space is a twisted torus, i.e. a compact space
constructed from identifying a group manifold under a discrete subgroup. This
incorporates reductions with duality twists, T-folds and a class of flux
compactifications, together with the non-geometric backgrounds expected to
arise from these through T-duality. It also incorporates backgrounds that are
not even locally geometric, and suggests a generalisation of T-duality to a
more general context. We discuss the effective field theory arising from such
an internal sector, give a world-sheet sigma model formulation of string theory
on such backgrounds and illustrate our discussion with detailed examples.
