Recently I attended a talk where the speaker introduced and used some notions of T-folds as non-geometric backgrounds. Although I had heard about doubled geometry, double field theory and about doing physics at both $T_p\mathcal{M}$ and $T_p^*\mathcal{M}$ I knew no details. One of the astonishing things I saw was that it is possible to construct some generalized metric that includes both $G_{\mu \nu}$ and $B_{\mu \nu}$. This metric is defined on some generalization of a manifold, the T-fold. **Ok, what is a T-fold? **

One thing I would also like to understand is how exactly one constructs this generalized metric from the two fields above? Also, once one has this generalized metric, is it used as a "generalized background"? I understand that if the B filed is turned of one recovers classical geometry but I struggle to understand at an intuitive level what this T-geometry really is. How does this construction help us to do physics and what physics one does in these generalized backgrounds?