We construct many examples of nonslice knots in 3-space that cannot be distinguished from slice knots by previously known invariants. Using Whitney towers in place of embedded disks, we define a geometric filtration of the 3-dimensional topological knot concordance group. The bottom part of the filtration exhibits all classical concordance invariants, including the Casson-Gordon invariants. As a first step, we construct an infinite sequence of new obstructions that vanish on slice knots. These take values in the $L$-theory of skew fields associated to certain universal groups. Finally, we use the dimension theory of von Neumann algebras to define an $L^2$-signature and use this to detect the first unknown step in our obstruction theory.