  In this paper, we study hypersurfaces in $\mathbb{H}^2\times\mathbb{H}^2$. We
first classify the hypersurfaces with constant principal curvatures and
constant product angle function. Then, we classify homogeneous hypersurfaces
and isoparametric hypersurfaces, respectively. Finally, we classify the
hypersurfaces with at most two distinct constant principal curvatures, as well
as those with three constant principal curvatures under some additional
conditions.
