$l^2$-betti numbers were defined by atiyah for compact manifolds x, by cheeger and gromov for countable groups $\gamma$ and by gaboriau for countable probability measure preserving equivalence relations. henrik d. petersen proposed a definition of $l^2$-betti numbers for locally compact groups. i will present this definition and a recent joint work with kyed and petersen in which we prove that the $l^2$-betti numbers of a locally compact group coincide with those of any associated cross section equivalence relation. as a consequence, we obtain several vanishing theorems for the reduced $l^2$-cohomology of locally compact groups.