  We study the geometry of stable maximal hypersurfaces in a variety of
spacetimes satisfying various physically relevant curvature assumptions, for
instance the Timelike Convergence Condition (TCC). We characterize stability
when the target space has constant sectional curvature as well as give
sufficient conditions on the geometry of the ambient spacetime (e.g., the
validity of TCC) to ensure stability. Some rigidity results and height
estimates are also proven in GRW spacetimes. In the last part of the paper we
consider $k$-stability of spacelike hypersurfaces, a concept related to mean
curvatures of higher orders.
