Introduction to the Farrell-Jones Conjectures

The conjecture in K-theory gives a description of the important and difficult to compute K-theory of the group ring Z[G] via the assembly map. This K-theory is crucial for classification of manifolds with fundamental group G. I will give a historical perspective on the conjecture and its further development to other theories and some more sophisticated reformulations. I will indicate an approach to the K-theoretic assembly map from the recent joint work with Gunnar Carlsson.