  The doubled formulation of string theory, which is T-duality covariant and
enlarges spacetime with extra coordinates conjugate to winding number, is
reformulated and its geometric and topological features examined. It is used to
formulate string theory in T-fold backgrounds with T-duality transition
functions and a quantum implementation of the constraints of the doubled
formalism is presented. This establishes the quantum equivalence to the usual
sigma-model formalism for world-sheets of arbitrary genus, provided a
topological term is added to the action. The quantisation involves a local
choice of polarisation, but the results are independent of this. The natural
dilaton of the doubled formalism is duality-invariant and so T-duality is a
perturbative symmetry for the perturbation theory in the corresponding coupling
constant. It is shown how this dilaton is related to the dilaton of the
conventional sigma-model which does transform under T-duality. The
generalisation of the doubled formalism to the superstring is given and shown
to be equivalent to the usual formulation. Finally, the formalism is
generalised to one in which the whole spacetime is doubled.
