  After we have given a survey on the Burnside ring of a finite group, we
discuss and analyze various extensions of this notion to infinite (discrete)
groups. The first three are the finite-G-set-version, the inverse-limit-version
and the covariant Burnside group. The most sophisticated one is the fourth
definition as the equivariant zero-th cohomotopy of the classifying space for
proper actions. In order to make sense of this definition we define equivariant
cohomotopy groups of finite proper equivariant CW-complexes in terms of maps
between the sphere bundles associated to equivariant vector bundles. We show
that this yields an equivariant cohomology theory with a multiplicative
structure. We formulate a version of the Segal Conjecture for infinite groups.
All this is analogous and related to the question what are the possible
extensions of the notion of the representation ring of a finite group to an
infinite group. Here possible candidates are projective class groups, Swan
groups and the equivariant topological K-theory of the classifying space for
proper actions.
