  We define for arbitrary modules over a finite von Neumann algebra $\cala$ a
dimension taking values in $[0,\infty]$ which extends the classical notion of
von Neumann dimension for finitely generated projective $\cala$-modules and
inherits all its useful properties such as additivity, cofinality and
continuity. This allows to define $L^2$-Betti numbers for arbitrary topological
spaces with an action of a discrete group $\Gamma$ extending the well-known
definition for regular coverings of compact manifolds. We show for an amenable
group $\Gamma$ that the $p$-th $L^2$-Betti number depends only on the
$\cc\Gamma$-module given by the $p$-th singular homology. Using the generalized
dimension function we detect elements in $G_0(\cc\Gamma)$, provided that
$\Gamma$ is amenable. We investigate the class of groups for which the zero-th
and first $L^2$-Betti numbers resp. all $L^2$-Betti numbers vanish. We study
$L^2$-Euler characteristics and introduce for a discrete group $\Gamma$ its
Burnside group extending the classical notions of Burnside ring and Burnside
ring congruences for finite $\Gamma$.
  Keywords: Dimension functions for finite von Neumann algebras, $L^2$-Betti
numbers, amenable groups, Grothendieck groups, Burnside groups
