  To determine the Lie groups that admit a flat (eventually complete) left
invariant semi-Riemannian metric is an open and difficult problem. The main aim
of this paper is the study of the flatness of left invariant semi Riemannian
metrics on quadratic Lie groups i.e. Lie groups endowed with a bi-invariant
semi Riemannian metric. We give a useful necessary and sufficient condition
that guaranties the flatness of a left invariant semi Riemannian metric defined
on a quadratic Lie group. All these semi Riemannian metrics are complete. We
show that there are no Riemannian or Lorentzian flat left invariant metrics on
non Abelian quadratic Lie groups, and that every quadratic 3 step nilpotent Lie
group admits a flat left invariant semi Riemannian metric. The case of
quadratic 2 step nilpotent Lie groups is also addressed.
