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We give a complete description of all hypersurfaces of the product spaces$\Sf^n\times \R$ and $\Hy^n\times \R$ that have flat normal bundle whenregarded as submanifolds with codimension two of the underlying flat spaces$\R^{n+2}\supset \Sf^n\times \R$ and $\Le^{n+2}\supset \Hy^n\times \R$. Weprove that any such hypersurface in $\Sf^n\times \R$ (respectively,$\Hy^n\times \R$) can be constructed by means of a family of parallelhypersurfaces in $\Sf^n$ (respectively, $\Hy^n$) and a smooth function of onevariable. Then we show that constant mean curvature hypersurfaces in this classare given in terms of an isoparametric family in the base space and a solutionof a certain ODE. For minimal hypersurfaces such solution is explicitlydetermined in terms of the mean curvature function of the isoparametric family.As another consequence of our general result, we classify the constant anglehypersurfaces of $\Sf^n\times \R$ and $\Hy^n\times \R$, that is, hypersurfaceswith the property that its unit normal vector field makes a constant angle withthe unit vector field spanning the second factor $\R$. This extends previousresults by Dillen, Fastenakels, Van der Veken, Vrancken and Munteanu forsurfaces in $\Sf^2\times \R$ and $\Hy^2\times \R$.  Our method also yields a classification of all Euclidean hypersurfaces withthe property that the tangent component of a constant vector field in theambient space is a principal direction, in particular of all Euclideanhypersurfaces whose unit normal vector field makes a constant angle with afixed direction.

On a class of hypersurfaces in $\Sf^n\times \R$ and $\Hy^n\times \R$