  We introduce a notion of $L^2$-Betti numbers for locally compact, second
countable, unimodular groups. We study the relation to the standard notion of
$L^2$-Betti numbers of countable discrete groups for lattices. In this way,
several new computations are obtained for countable groups, including lattices
in algebraic groups over local fields, and Kac-Moody lattices.
  We also extend the vanishing of reduced $L^2$-cohomology for countable
amenable groups, a well known theorem due to Cheeger and Gromov, to cover all
amenable, second countable, unimodular locally compact groups.
