In noncommutative geometry (NCG) one typically treats unital C*-algebras as noncommutative compact spaces via Gelfand-Naimark duality. In various applications of NCG to problems in geometry or topology it is customary to first reformulate these problems in terms of certain (co)homology theories for noncommutative spaces. The celebrated Baum-Connes conjecture, that reduces the Novikov conjecture to an assertion in bivariant K-theory, is a prime example of this principle. However, the category of C*-algebras is well-known to be de ficient from the viewpoint of homotopy theory or index theory. In this talk I am going to first survey certain (co)homology theories for noncommutative spaces, then present my proposed solution to the aforementioned problem, and finally (time permitting) discuss some applications. I will try to keep it non-technical and accessible to a wide range of audience.

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