  Let $(G,\Omega)$ be a symplectic Lie group, i.e, a Lie group endowed with a
left invariant symplectic form. If $\G$ is the Lie algebra of $G$ then we call
$(\G,\omega=\Om(e))$ a symplectic Lie algebra. The product $\bullet$ on $\G$
defined by $3\omega\left(x\bullet
y,z\right)=\omega\left([x,y],z\right)+\omega\left([x,z],y\right)$ extends to a
left invariant connection $\na$ on $G$ which is torsion free and symplectic
($\na\Om=0)$. When $\na$ has vanishing curvature, we call $(G,\Omega)$ a flat
symplectic Lie group and $(\G,\om)$ a flat symplectic Lie algebra. In this
paper, we study flat symplectic Lie groups. We start by showing that the
derived ideal of a flat symplectic Lie algebra is degenerate with respect to
$\om$. We show that a flat symplectic Lie group must be nilpotent with
degenerate center. This implies that the connection $\na$ of a flat symplectic
Lie group is always complete. We prove that the double extension process can be
applied to characterize all flat symplectic Lie algebras. More precisely, we
show that every flat symplectic Lie algebra is obtained by a sequence of double
extension of flat symplectic Lie algebras starting from $\{0\}$. As examples in
low dimensions, we classify all flat symplectic Lie algebras of dimension
$\leq6$.
