Survey on Classifying Spaces for Families of Subgroups

We define for a topological group G and a family of subgroups $$ \mathcal{F} $$ two versions for the classifying space for the family $$ \mathcal{F} $$ , the G-CW-version $$ E_\mathcal{F} $$ (G) and the numerable G-space version $$ J_\mathcal{F} $$ (G). They agree if G is discrete, or if G is a Lie group and each element in $$ \mathcal{F} $$ compact, or if $$ \mathcal{F} $$ 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.