  This thesis investigates the quantum properties of T-duality invariant
formalisms of String Theory. We introduce and review duality invariant
formalisms of String Theory including the Doubled Formalism. We calculate the
background field equations for the Doubled Formalism of Abelian T-duality and
show how they are consistent with those of a conventional String Theory
description of a toroidal compactification. We generalise these considerations
to the case of Poisson--Lie T-duality and show that the system of
renormalisation group equations obtained from the duality invariant parent
theory are equivalent to those of either of the T-dual pair of sigma-models. In
duality invariant formalisms it is quite common to loose manifest Lorentz
invariance at the level of the Lagrangian. The lack of manifest invariance
means that at the quantum level one might anticipate Lorentz anomalies and we
show that such anomalies cancel non-trivially. These represent important and
non-trivial consistency checks of the duality invariant approach to String
Theory.
