  We define for a topological group G and a family of subgroups F two versions
for the classifying space for the family F, the G-CW-version E_F(G) and the
numerable G-space version J_F(G). They agree if G is discrete, or if G is a Lie
group and each element in F compact, or if F is the family of compact
subgroups. We discuss special geometric models for these spaces for the family
of compact open groups in special cases such as almost connected groups G and
word hyperbolic groups G. We deal with the question whether there are finite
models, models of finite type, finite dimensional models. We also discuss the
relevance of these spaces for the Baum-Connes Conjecture about the topological
K-theory of the reduced group C^*-algebra, for the Farrell-Jones Conjecture
about the algebraic K- and L-theory of group rings, for Completion Theorems and
for classifying spaces for equivariant vector bundles and for other situations.
