  Let G be a discrete group. We give methods to compute for a generalized
(co-)homology theory its values on the Borel construction (EG x X)/G of a
proper G-CW-complex X satisfying certain finiteness conditions. In particular
we give formulas computing the topological K-(co)homology of the classifying
space BG up to finite abelian torsion groups. They apply for instance to
arithmetic groups, word hyperbolic groups, mapping class groups and discrete
cocompact subgroups of almost connected Lie groups. For finite groups G these
formulas are sharp. The main new tools we use for the K-theory calculation are
a Cocompletion Theorem and Equivariant Universal Coefficient Theorems which are
of independent interest. In the case where G is a finite group these theorems
reduce to well-known results of Greenlees and Boekstedt.
