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We present a ten-dimensional theory, named \beta-supergravity, that containsnon-geometric fluxes and could uplift some four-dimensional gaugedsupergravities. Building on earlier work, we study here its NSNS sector, whereQ- and R-fluxes are precisely identified. Interestingly, the Q-flux is capturedin an analogue of the Levi-Civita spin connection, giving rise to a secondcurvature scalar. We reproduce the ten-dimensional Lagrangian using theGeneralized Geometry formalism; this provides us with enlighteninginterpretations of the new structures. Then, we derive the equations of motionof our theory, and finally discuss further aspects: the dimensional reductionto four dimensions and comparison to gauged supergravities, the obtention often-dimensional purely NSNS solutions, the extensions to other sectors and newobjects, the supergravity limit, and eventually the symmetries, in particularthe \beta gauge transformation. We also introduce the related notion of ageneralized cotangent bundle.

\beta-supergravity: a ten-dimensional theory with non-geometric fluxes, and its geometric framework