  We show that the Kauffman bracket skein module of a closed Seifert fibered
3-manifold $M$ is finitely generated over $\mathbb Z[A^{\pm 1}]$ if and only if
$M$ is irreducible and non-Haken.
  We analyze in detail the character varieties $X(M)$ of such manifolds and
show that under mild conditions they are reduced. We compute the Kauffman
bracket skein modules for these $3$-manifolds (over $\mathbb Q(A)$) and show
that their dimensions coincide with $|X(M)|.$
