$L^2$-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.