An interesting problem in submanifold theory is to study
isometric immersions $f : M^n \to \mathbb{Q}_c^{n+p}$ of a connected complete 
Riemannian manifold of dimension $n$ acted on by a closed connected 
subgroup of its isometry group $Iso(M^n)$. This study was initiated by 
Kobayashi in 1958, who proved that if $c=0$, $p=1$, and $M^n$ is 
compact and homogeneous, i.e., $Iso(M^n)$ acts transitively on $M^n$, 
then $f(M^n)$ must be a round sphere. In this talk we will considerer
hypersurfaces $f : M^n \to \mathbb{Q}_c^{n+1}$ of a complete Riemannian 
manifold $M^n$ on which a compact, connected subgroup $G$ of
$Iso(M^n)$ acts with maximal dimensional orbits of codimension one. 
We call $f$ a hypersurface of $G$-cohomogeneity one. More
precisely, we will review some results about Euclidean hypersurfaces
and discuss a more recent work on such hypersurfaces in the 
hyperbolic space, where we provide a characterization if either $n \geq 3$
and $M^n$ is compact, or $n \geq 5$ and the connected components 
of the set where the sectional curvature is constant and equal to $-1$ 
are bounded.