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In this paper, we classify the hypersurfaces in $\mathbb{S}^{n}\times\mathbb{R}$ and $\mathbb{H}^{n}\times\mathbb{R}$, $n\neq 3$, with $g$ distinctconstant principal curvatures, $g\in\{1,2,3\}$, where $\mathbb{S}^{n}$ and$\mathbb{H}^{n}$ denote the sphere and hyperbolic space of dimension $n$,respectively. We prove that such hypersurfaces are isoparametric in thosespaces. Furthermore, we find a necessary and sufficient condition for anisoparametric hypersurface in $\mathbb{S}^{n}\times \mathbb{R}\subset\mathbb{R}^{n+2}$ and $\mathbb{H}^{n}\times\mathbb{R}\subset \mathbb{L}^{n+2}$with flat normal bundle, having constant principal curvatures.

Hypersurfaces with constant principal curvatures in $\mathbb{S}^{n}\times\mathbb{R}$ and $\mathbb{H}^{n}\times\mathbb{R}$