### Duality Symmetric String and M-Theory

We review recent developments in duality symmetric string theory.
We begin with the world sheet doubled formalism which describes
strings in an extended space time with extra coordinates conjugate
to winding modes. This formalism is T-duality symmetric and can
accommodate non-geometric T-fold backgrounds which are beyond the
scope of Riemannian geometry. Vanishing of the conformal anomaly of
this theory can be interpreted as a set of spacetime equations for
the background fields. These equations follow from an action
principle that has been dubbed Double Field Theory (DFT). We review
the aspects of generalised geometry relevant for DFT. We outline
recent extensions of DFT and explain how, by relaxing the so-called
strong constraint with a Scherk Schwarz ansatz, one can obtain
backgrounds that simultaneously depend on both the regular and
T-dual coordinates. This provides a purely geometric higher
dimensional origin to gauged supergravities that arise from
non-geometric compactification. We then turn to M-theory and
describe recent progress in formulating an \(E_{n(n)}\) U-duality
covariant description of the dynamics. We describe how spacetime
may be extended to accommodate coordinates conjugate to brane
wrapping modes and the construction of generalised metrics in this
extend space that unite the bosonic fields of supergravity into a
single object. We review the action principles for these theories
and their novel gauge symmetries. We also describe how a Scherk
Schwarz reduction can be applied in the M-theory context and the
resulting relationship to the embedding tensor formulation of
maximal gauged supergravities.