  The main aim of this paper is the description of a large class of lattices in
some nilpotent Lie groups, sometimes filiformes, carrying a flat left invariant
linear connection anf often a left invariant symplectic form. As a consequence
we obtain an infinity of, non homeomorphic, compact affine or symplectic
manifolds. We review some new facts about the geometry of compact symplectic
nilmanifolds and we describe symplectic reduction for these manifolds. For the
Heisenberg Lie group, defined over a local associative and commutative finite
dimensional real algebra, a necessary and suffisant condition for the existence
of a left invarian symplectic form, is given. In the symplectic case we show a
that a lattice in the group determines naturally lattices in the double Lie
group corresponding to any solution of the classical Yang-Baxter equation.
