  We investigate in detail solutions of supergravity that involve warped
products of flat geometries of the type M(p+1) x R x T(D-p-2) depending on a
single coordinate. In the absence of fluxes, the solutions include flat space
and Kasner-like vacua that break all supersymmetries. In the presence of a
symmetric flux, there are three families of solutions that are characterized by
a pair of boundaries and have a singularity at one of them, the origin. The
first family comprises supersymmetric vacua, which capture a universal limiting
behavior at the origin. The first and second families also contain
non--supersymmetric solutions whose behavior at the other boundary, which can
lie at a finite or infinite distance, is captured by the no--flux solutions.
The solutions of the third family have a second boundary at a finite distance
where they approach again the supersymmetric backgrounds. These vacua exhibit a
variety of interesting scenarios, which include compactifications on finite
intervals and (p+1)-dimensional effective theories where the string coupling
has an upper bound. We also build corresponding cosmologies, and in some of
them the string coupling can be finite throughout the evolution.
