This talk will be mainly concerned with a non-standard 'algebraic topology' built on knots. In particular, we will define {\it the Kauffman bracket skein module} $S(M)$, a module associated to any 3-dimensional manifold $M$. A particular version of this module has the structure of a commutative algebra and therefore it is called a {\it skein algebra}. The skein algebra of a manifold depends only on its fundamental group. Motivated by this, we generalize the notion of the skein algebra of a manifold to that of a skein algebra of a group.

We show that the skein algebra of the group, $G$, is, in fact, an algebra associated to $Sl_2(C)$-representations of $G$ defined and investigated by Brumfiel and Hilden. This result implies that the skein algebra of $G$ is, up to nilpotent elements, a coordinate ring of a {\it character variety} of $G$, an algebraic set representing all traces of homomorphisms of $G$ into $Sl_2(C)$.

We are going to enumerate classes of groups for which skein algebras do not have nilpotent elements.