  In this paper a large class of Fano double quadrics and cubics are shown to
be factorial and birationally superrigid, in particular they admit no
non-trivial structure of a fibration with rationally connected fibres and are
therefore non-rational. This is shown using the "Method of maximal
singularities" of Iskovskikh and Manin, expanded upon by Pukhlikov. In
addition, an estimate of the codimension of the set of such varieties is
calculated.
