  We present a ten-dimensional theory, named \beta-supergravity, that contains
non-geometric fluxes and could uplift some four-dimensional gauged
supergravities. Building on earlier work, we study here its NSNS sector, where
Q- and R-fluxes are precisely identified. Interestingly, the Q-flux is captured
in an analogue of the Levi-Civita spin connection, giving rise to a second
curvature scalar. We reproduce the ten-dimensional Lagrangian using the
Generalized Geometry formalism; this provides us with enlightening
interpretations of the new structures. Then, we derive the equations of motion
of our theory, and finally discuss further aspects: the dimensional reduction
to four dimensions and comparison to gauged supergravities, the obtention of
ten-dimensional purely NSNS solutions, the extensions to other sectors and new
objects, the supergravity limit, and eventually the symmetries, in particular
the \beta gauge transformation. We also introduce the related notion of a
generalized cotangent bundle.
