  A hypersurface is said to be totally biharmonic if all its geodesics are
biharmonic curves in the ambient space. We prove that a totally biharmonic
hypersurface into a space form is an isoparametric biharmonic hypersurface,
which allows us to give the full classification of totally biharmonic
hypersurfaces in these spaces. Moreover, restricting ourselves to the
3-dimensional case, we show that totally biharmonic surfaces into
Bianchi-Cartan-Vranceanu spaces are isoparametric surfaces and we give their
full classification. In particular, we show that, leaving aside surfaces in the
3-dimensional sphere, the only non-trivial example of a totally biharmonic
surface appears in the product space $\mathbb{S}^2(\rho)\times\mathbb{R}$.
