  For certain classes of knots we define geometric invariants called
higher-order genera. Each of these invariants is a refinement of the slice
genus of a knot. We find lower bounds for the higher-order genera in terms of
certain von Neumann $\rho$-invariants, which we call higher-order signatures.
The higher-order genera offer a refinement of the Grope filtration of the knot
concordance group.
