One of the ways in which string theory differs from conventional field
theories is that it has duality symmetries, which allow the construction of
so-called non-geometric backgrounds, such as T-folds which have T-duality
transition functions. String theory on a torus requires the introduction of
dual coordinates conjugate to string winding number. This leads to physics
and novel geometry in a doubled space, with non-trivial dynamics in the full
doubled space-time. The geometry and physics of doubled space-time will be
developed and discussed, along with generalisations to M-theory.