We introduce a notion of L2-Betti numbers for locally compact, second countable, unimodular groups. We study the relation to the standard notion of L2-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 elds, and Kac-Moody lattices. We also extend the vanishing of reduced L2-cohomology for countable amenable groups, a well known theorem due to Cheeger and Gromov, to cover all amenable, second countable, unimodular locally compact groups.