  Double field theory and exceptional field theory are formulations of
supergravity that make certain dualities manifest symmetries of the action. To
achieve this, the geometry is extended by including dual coordinates
corresponding to winding modes of the fundamental objects. This geometrically
unifies the spacetime metric and the gauge fields (and their local symmetries)
in a generalized geometry. Solutions to these extended field theories take the
simple form of waves and monopoles in the extended space. From a supergravity
point of view they appear as 1/2 BPS objects such as the string, the membrane
and the fivebrane in ordinary spacetime. In this thesis double field theory and
exceptional field theory are introduced, solutions to their equations of motion
are constructed and their properties are analyzed. Further it is established
how isometries in the extended space give rise to duality relations between the
supergravity solutions. Extensions to these core ideas include studying
Goldstone modes, probing singularities at the core of solutions and localizing
them in winding space. The relation of exceptional field theory to F-theory is
also covered providing an action for the latter and incorporating the duality
between M-theory and F-theory.
