Eddy-current inversion is the process whereby the geometry of a flaw in a metal is derived from electromagnetic probe measurements. An inversion scheme is described for finding the shape and size of cracks from eddy-current probe impedance measurements. The approach is based on an optimization scheme that seeks to minimize a global error function quantifying the difference between predicted and observed probe impedances. The error minimum is sought using a standard descent algorithm that requires a knowledge of the gradient of the error with respect to a variation of the flaw geometry. Computation of the gradient is based on a provisional flaw estimate, then the flaw estimate is updated in a “direction” that reduces the error. The process continues iteratively until a convergence criterion has been satisfied. Then the final flaw estimate should match the shape of the real defect. An equation for the gradient has been derived using an integral formulation of the ideal crack problem. Numerical estimates of the error gradient and the probe impedances have been calculated using approximations based on the moment method. Tests of the inversion scheme using single frequency probe impedance measurements have been carried out by calculating the shapes of narrow slots in aluminum alloy plates. Good agreement is found between the optimum profiles and the measured slot shapes.