  We construct many examples of non-slice 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 {\em 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.
