  This paper deals essentially with affine or projective transformations of Lie
groups endowed with a flat left invariant affine or projective structure. These
groups are called flat affine or flat projective Lie groups. Our main results
determine Lie groups admitting flat bi-invariant affine or projective
structures. These groups could play an essential role in the study of
homogeneous spaces $M=G/H$ admitting flat affine or flat projective structures
invariant under the natural action of $G$ on $M$. A. Medina asked several years
ago if the group of affine transformations of a flat affine Lie group is a flat
projective Lie group. In this work we provide a partial possitive answer to
this question.
