Introduction. There is a general question as to how much can be said about a filtered object through the knowledge of its associated graded object. We consider here a particular case of this general problem. We take the symmetric algebra S(L) of a free K-module L and look for filtered K-algebras whose associated graded algebras are isomorphic to S(L). Some such algebras are already known. In fact if g denotes an arbitrary Lie algebra on L, the "Poincar6-Witt Theorem" asserts that the universal enveloping algebra of g is one such. It turns out that this gives "almost" a general solution of our problem. Indeed, the algebras we seek are suitable generalizations of the usual enveloping algebras and can in fact be defined as universal objects for certain "generalized representations" of Lie algebras on L. The rest of our results are on the cohomology of these algebras. For a Lie algebra g over a commutative ring K and a 2-cocycle f on its standard complex with values in K, we define in ?1 the notion of an frepresentation. The usual representations of g correspond to the case f= 0. We introduce in ?2 the filtered K-algebra gf which is a universal model for f-representations. We deduce the "Poincare-Witt Theorem" (Theorem 2.6) for gf as an easy consequence of the usual Poincare-Witt Theorem, proved in [1, p. 271]. It is then clear that if g is K-free, there is a graded K-algebra isomorphism 4t'f: S(g) -*E0(gf), where S(g) denotes the symmetric algebra of the K-module g and EO(gf) the graded algebra associated with gf (Theorem 2.5). In ?3, we define, for a fixed graded K-algebra S, the category whose objects are pairs (A, 41A), where A is a filtered K-algebra, IPA: SEO(A) an isomorphism of graded algebras and whose maps are defined in an obvious manner. If S is the symmetric algebra of a free K-module L, then there is a 1-1 correspondence between isomorphism classes of such objects and pairs (g, f), where g is a Lie algebra on L and f H2(g, K). For a cocyclef in the class f, the pair (gf, ifr) is an object in the class corresponding to (g, f) (Theorem 3.1). The fourth section is devoted to the computations of certain of the usual homology and cohomology groups of a finite dimensional Lie algebra g. This amounts to a study of g& for f= 0. These computations are used in the next