Spaces over a category and assembly maps in isomorphism conjectures in K- and L-theory

We give a unified approach to the Isomorphism Conjecture of Farrell and Jones on the algebraic K- and L-theory of integral group rings and to the Baum-Connes Conjecture on the topological K-theory of reduced C*-algebras of groups. The approach is through spectra over the orbit category of a discrete group G. We give several points of view on the assembly map for a family of subgroups and characterize such assembly maps by a universal property generalizing the results of Weiss and Williams to the equivariant setting. The main tools are spaces and spectra over a category and their associated generalized homology and cohomology theories, and homotopy limits.