  The gauged sigma-model argument that string backgrounds related by T-dual
give equivalent quantum theories is revisited, taking careful account of global
considerations. The topological obstructions to gauging sigma-models give rise
to obstructions to T-duality, but these are milder than those for gauging: it
is possible to T-dualise a large class of sigma-models that cannot be gauged.
For backgrounds that are torus fibrations, it is expected that T-duality can be
applied fibrewise in the general case in which there are no globally-defined
Killing vector fields, so that there is no isometry symmetry that can be
gauged; the derivation of T-duality is extended to this case. The T-duality
transformations are presented in terms of globally-defined quantities. The
generalisation to non-geometric string backgrounds is discussed, the conditions
for the T-dual background to be geometric found and the topology of T-folds
analysed.
