  This paper deals with affine connections on real manifolds. We give a new
characterization of flat affine connections on real manifolds by means of
certain affine representations of the Lie group of automorphisms preserving the
connection. Then we specialize the characterization to the case of a left
invariant connection on a Lie group. In the last case, we show the existence of
a Lie group endowed with a flat affine bi-invariant connection whose Lie
algebra contains the Lie algebra of complete infinitesimal affine
transformations of the given Lie group. We also prove some results about flat
affine manifolds whose group of diffeomorphisms admit a flat affine
bi-invariant structure. The paper is illustrated with several examples.
