  A birational map from a projective space onto a not too much singular
projective variety with a single irreducible non-singular base locus scheme
(special birational transformation) is a rare enough phenomenon to allow
meaningful and concise classification results.
  We shall concentrate on transformations defined by quadratic equations onto
some varieties (especially projective hypersurfaces of small degree), where
quite surprisingly the base loci are interesting projective manifolds appearing
in other contexts; for example, exceptions for adjunction theory, small degree
or small codimensional manifolds, Severi or more generally homogeneous
varieties.
  In particular, we shall classify: quadro-quadric transformations into a
quadric hypersurface; quadro-cubic transformations into a del Pezzo variety;
transformations whose base locus (scheme) has dimension at most three.
