  A manifestly T-dual invariant formulation of bosonic string theory is
discussed here. It can be obtained by making both the usual string compact
coordinates and their duals explicitly appear, on the same footing, in the
world-sheet action. A peculiarity of such a model is the loss of the local
Lorentz invariance which is required to be recovered on-shell. This dictates a
constraint on the backgrounds which characterizes the double geometry of the
target space. Constant and non-constant backgrounds are considered. In the
former case, the local Lorentz constraint implies the geometry of a double
torus with an $O(D,D)$ invariance. In the latter, it is shown how and when the
$O(D,D)$ invariance still holds and when deformations from it can be implied.
Results of the quantization of the free theory are also briefly exhibited.
