  Special birational transformations $\Phi:\p^r\da Z$ defined by quadric
hypersurfaces are studied by means of the variety of lines $\mathcal
L_z\subset\p^{r-1}$ passing through a general point $z\in Z$. Classification
results are obtained when $Z$ is either a Grassmannian of lines, or the
10-dimensional spinor variety, or the $E_6$-variety. In the particular case of
quadro-quadric transformations, we extend the well-known classification of Ein
and Shepherd-Barron coming from Zak's classification of Severi varieties to a
wider class of prime Fano manifolds $Z$. Combining both results, we get a
classification of special birational transformations $\Phi:\p^r\da Z$ defined
by quadric hypersurfaces onto (a linear setion of) a rational homogeneous
variety different from a projective space and a quadric hypersurface.
