  We give a characterization of flat affine connections on manifolds by means
of a natural affine representation of the universal covering of the Lie group
of diffeomorphisms preserving the connection. From the infinitesimal point of
view, this representation is determined by the 1-connection form and the
fundamental form of the bundle of linear frames of the manifold. We show that
the group of affine transformations of a real flat affine $n$-dimensional
manifold, acts on $\mathbb{R}^n$ leaving an open orbit when its dimension is
greater than $n$. Moreover, when the dimension of the group of affine
transformations is $n$, this orbit has discrete isotropy. For any given Lie
subgroup $H$ of affine transformations of the manifold, we show the existence
of an associative envelope of the Lie algebra of $H$, relative to the
connection. The case when $M$ is a Lie group and $H$ acts on $G$ by left
translations is particularly interesting. We also exhibit some results about
flat affine manifolds whose group of affine transformations admits a flat
affine bi-invariant structure. The paper is illustrated with several examples.
