  We define a set of "second-order" L^(2)-signature invariants for any
algebraically slice knot. These obstruct a knot's being a slice knot and
generalize Casson-Gordon invariants, which we consider to be "first-order
signatures". As one application we prove: If K is a genus one slice knot then,
on any genus one Seifert surface, there exists a homologically essential simple
closed curve of self-linking zero, which has vanishing zero-th order signature
and a vanishing first-order signature. This extends theorems of Cooper and
Gilmer. We introduce a geometric notion, that of a derivative of a knot with
respect to a metabolizer. We also introduce a new equivalence relation,
generalizing homology cobordism, called null-bordism.
