  We consider hypersurfaces of products $M\times\mathbb R$ with constant $r$-th
mean curvature $H_r\ge 0$ (to be called $H_r$-hypersurfaces), where $M$ is an
arbitrary Riemannian $n$-manifold. We develop a general method for constructing
them, and employ it to produce many examples for a variety of manifolds $M,$
including all simply connected space forms and the hyperbolic spaces
$\mathbb{H}_{\mathbb F}^m$ (rank $1$ symmetric spaces of noncompact type). We
construct and classify complete rotational $H_r(\ge 0)$-hypersurfaces in
$\mathbb{H}_{\mathbb F}^m\times\mathbb R$ and in $\mathbb S^n\times\mathbb R$
as well. They include spheres, Delaunay-type annuli and, in the case of
$\mathbb{H}_{\mathbb F}^m\times\mathbb R,$ entire graphs. We also construct and
classify complete $H_r(\ge 0)$-hypersurfaces of $\mathbb{H}_{\mathbb
F}^m\times\mathbb R$ which are invariant by either parabolic isometries or
hyperbolic translations. We establish a Jellett-Liebmann-type theorem by
showing that a compact, connected and strictly convex $H_r$-hypersurface of
$\mathbb H^n\times\mathbb R$ or $\mathbb S^n\times\mathbb R$ $(n\ge 3)$ is a
rotational embedded sphere. Other uniqueness results for complete
$H_r$-hypersurfaces of these ambient spaces are obtained.
