The Novikov conjecture is an important problem in higher dimensional topology. It claims that the higher signatures of a compact smooth manifold are invariant under orientation preserving homotopy equivalences. The Novikov conjecture is a consequence of the strong Novikov conjecture in the computation of the K-theory of group C*-algebras. In this talk, I will talk about the Novikov conjecture for groups which are extensions of coarsely embeddable groups. In particular, I will talk about the computation of the K-theory of some C*-algebras.