  We classify the homogeneous and isoparametric hypersurfaces of
$\mathbb{S}^2\times\mathbb{S}^2$. In the classification, besides the
hypersurfaces $\mathbb{S}^1(r)\times\mathbb{S}^2,\,r\in (0,1]$, it appears a
family of hypersurfaces with three different constant principal curvatures and
zero Gauss-Kronecker curvature. Also we classify the hypersurfaces of
$\mathbb{S}^2\times\mathbb{S}^2$ with at most two constant principal curvatures
and, under certain conditions, with three constant principal curvatures.
