  Geodesically complete affine manifolds are quotients of the Euclidean space
through a properly discontinuous action of a subgroup of affine Euclidean
transformations. An equivalent definition is that the tangent bundle of such a
manifold admits a flat, symmetric and complete connection. If the completeness
assumption is dropped, the manifold is not necessarily obtained as the quotient
of the Euclidean space through a properly discontinuous group of affine
transformations. In fact the universal cover may no longer be the Euclidean
space. The main result of this paper states that all compact affine manifolds
have 0 Euler characteristic and that the fundamental group of these manifolds
is non-trivial.
