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In this article spacelike hypersurfaces immersed in twisted productspacetimes $I\times_f F$ with complete fiber are studied. Several conditionsensuring global hyperbolicity are presented, as well as a relation that needsto hold on each spacelike hypersurface in $I\times_f F$ for it to be a simplewarped product. When the fiber is assumed to be closed (compact and withoutboundary) and the ambient spacetime has a suitable expanding behaviour,non-existence results for constant mean curvature hypersurfaces are obtained.Under the same hypothesis, a characterization of compact maximal hypersurfacesand other for totally umbilic ones with a suitable restriction on their meancurvature are presented. The description of maximal hypersurfaces in twistedproduct spacetimes of the form $I\,{ }_{f}\!\!\times F$ with a one-dimensionalLorentzian fiber is also included. Finally, the mean curvature equation for aspacelike graph on the fiber is computed and as an application, someCalabi-Bernstein-type results are proven. We also include in an Appendix someknown conformal geometry results describing the transformation of relevanttensors and operators under the action of a conformal map in apseudo-Riemannian background.

Spacelike hypersurfaces in twisted product spacetimes with complete fiber and Calabi-Bernstein-type problems