  This paper is devoted to study the generic fold-fold singularity of Filippov
systems on the plane, its unfoldings and its Sotomayor-Teixeira regularization.
We work with general Filippov systems and provide the bifurcation diagrams of
the fold-fold singularity and their unfoldings, proving that, under some
generic conditions, is a codimension one embedded submanifold of the set of all
Filippov systems. The regularization of this singularity is studied and its
bifurcation diagram is shown. In the visible-invisible case, the use of
geometric singular perturbation theory has been useful to give the complete
diagram of the unfolding, specially the appearance and disappearance of
periodic orbits that are not present in the Filippov vector field. In the case
of a linear regularization, we prove that the regularized system is equivalent
to a general slow-fast system studied by Krupa and Szmolyan [KrupaS01].
