  We consider compactifications of type II string theory in which a
d-dimensional torus is fibered over a base X. In string theory, the transition
functions of this fibration need not be simply diffeomorphisms of T^d but can
involve elements of the T-duality group Spin(d,d,Z). We precisely define the
notion of a T-fold with NSNS flux. Given such a T-fold, we define the
Z_2-graded cohomology theory describing the unquantized RR field strengths and
discuss how the data of a T-fold can be interpreted in terms of generalized
NSNS fluxes and the twisted differential of Shelton-Taylor-Wecht.
