Double field theory and stringy geometry

<!--HTML-->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.