  In this article spacelike hypersurfaces immersed in twisted product
spacetimes $I\times_f F$ with complete fiber are studied. Several conditions
ensuring global hyperbolicity are presented, as well as a relation that needs
to hold on each spacelike hypersurface in $I\times_f F$ for it to be a simple
warped product. When the fiber is assumed to be closed (compact and without
boundary) 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 hypersurfaces
and other for totally umbilic ones with a suitable restriction on their mean
curvature are presented. The description of maximal hypersurfaces in twisted
product spacetimes of the form $I\,{ }_{f}\!\!\times F$ with a one-dimensional
Lorentzian fiber is also included. Finally, the mean curvature equation for a
spacelike graph on the fiber is computed and as an application, some
Calabi-Bernstein-type results are proven. We also include in an Appendix some
known conformal geometry results describing the transformation of relevant
tensors and operators under the action of a conformal map in a
pseudo-Riemannian background.
