  Let X be a smooth projective variety of dimension n on which a simple Lie
group G acts regularly and non trivially. Then X is not minimal in the sense of
the Minimal Model Program. In the paper we work out a classification of X via
the Minimal Model Program under the assumption that the dimension of X is small
with the respect to the dimension of G. More precisely we classify all such X
with n smaller or equal to (r_G +1), where r_G is the minimum codimension of
the maximal parabolic subgroup of G (for instance r_{SL(m)}= m-1). We consider
also the case when G = SL(3) and X is a smooth 4-fold on which G acts with an
open orbit.
