Summary: 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 one 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.