  Let \phi: \mathbb{P}^{r}\dashrightarrow Z be a birational transformation with
a smooth connected base locus scheme, where Z\subseteq\mathbb{P}^{r+c} is a
nondegenerate prime Fano manifold. We call \phi a quadro-quadric special
briational transformation if \phi and \phi^{-1} are defined by linear
subsystems of |\mathcal{O}_{\mathbb{P}^{r}}(2)| and |\mathcal{O}_{Z}(2)|
respectively. In this paper we classify quadro-quadric special birational
transformations in the cases where either (i) Z is a complete intersection and
the base locus scheme of \phi^{-1} is smooth, or (ii) Z is a hypersurface.
