  In this thesis the recently developed duality covariant approach to string
and M-theory is investigated. In this formalism the U-duality symmetry of
M-theory or T-duality symmetry of Type II string theory becomes manifest upon
extending coordinates that describe a background.
  The effective potential of Double Field Theory is formulated only up to a
boundary term and thus does not capture possible topological effects that may
come from a boundary. By introducing a generalised normal we derive a
manifestly duality covariant boundary term that reproduces the known
Gibbons-Hawking action of General Relativity, if the section condition is
imposed. It is shown that the full potential can be represented as a sum of the
scalar potential of gauged supergravity and a topological term that is a full
derivative. The latter is conjectured to capture non-trivial topological
information of the corresponding background, such as monodromy around an exotic
brane.
  Next we show that the Scherk-Schwarz reduction of M-theory extended geometry
successfully reproduces known structures of maximal gauged supergravities.
Local symmetries of the extended space defined by a generalised Lie derivatives
reduce to gauge transformations and lead to the embedding tensor written in
terms of twist matrices. The scalar potential of maximal gauged supergravity
that follows from the effective potential is shown to be duality invariant with
no need of section condition. Instead, this condition, that assures the closure
of the algebra of generalised diffeomorphisms, takes the form of the quadratic
constraints on the embedding tensor.
