  We call the Lie algebra of a Lie group with a left invariant
pseudo-Riemannian flat metric pseudo-Riemannian flat Lie algebra. We give a new
proof of a classical result of Milnor on Riemannian flat Lie algebras. We
reduce the study of Lorentzian flat Lie algebras to those with trivial center
or those with degenerate center. We show that the double extension process can
be used to construct all Lorentzian flat Lie algebras with degenerate center
generalizing a result of Aubert-Medina on Lorentzian flat nilpotent Lie
algebras. Finally, we give the list of Lorentzian flat Lie algebras with
degenerate center up to dimension 6.
