Abstract: A brief overview of calculating L^2-Betti numbers will be given. Then we shall determine the L^2-Betti numbers for a one-relator group. We shall use the techniques to show, for example, that for a left ordered two-generated group, either the group is free or the first L^2-Betti number is 0. If there is time, L^2-Betti numbers of one-relator surface groups will be considered.