  We discuss the consequences of the Poincar\'e duality, versus AS- Gorenstein
property, for Koszul algebras (homogeneous and non homogeneous). For
homogeneous Koszul algebras, the Poincar\'e duality property implies the
existence of twisted potentials which characterize the corresponding algebras
while in the case of quadratic linear Koszul algebras, the Poincar\'e duality
is needed to get a good generalization of universal enveloping algebras of Lie
algebras. In the latter case we describe and discuss the corresponding
generalization of Lie algebras. We also give a short review of the notion of
Koszulity and of the Koszul duality for N-homogeneous algebras and for the
corresponding nonhomogeneous versions.
