Non-destructive evaluation is concerned with the detection and characterisation of flaws. In eddy-current inspection, measurements of the electro-magnetic field are made at the surface of a flawed conductor, in an oscillating electric or magnetic field. The aim of this study is to develop an eddy-current method of imaging flaws based on measurements of the magnetic field data. In the presence of an oscillating current, it is shown that the physical behaviour of a thin crack can be modelled by a layer of electric dipoles. Green's functions are introduced, relating the dipole distribution on the crack to the magnetic field above the conductor. It is shown how the resulting integral expressions can predict the magnetic field. These relationships also provide a means of deducing the dipole density on a crack from measurements of the magnetic field. The distribution of dipoles reflects the crack geometry and essentially represents an image of the crack. A scanning rig is described and optimised for use in the study. Alternating current, at a pre-selected frequency (typically around 1kHz), is injected into aluminium alloy plates containing manufactured slots, and spatially distributed measurements of the magnetic field are made with a Hall transducer. The Hall transducer has advantages over traditional coil type transducers, in particular having better spatial resolution and improved detection efficiency, over the range of frequencies used in this study. The dipole density is generally a vector quantity and the problem of recovering it is ill-posed. However, by orientating the plane of the slot perpendicular to the direction of a uniform incident field, it was possible to induce only one component of the dipole density and then recover this from field measurements. The data collected by the experimental rig was used to image two types of manufactured slot. Initially, relatively shallow surface-breaking slots were imaged. For surface cracks, the imaging process was formulated as one of deconvolution, and a Wiener filter was used to perform this operation, to overcome the effects of experimental noise. Secondly, images were obtained of thin slots, which did not break the surface of the conductor, and whose plane extended into the specimens. For these sub-surface slots, the imaging problem was formulated as one of matrix inversion and several techniques were tested for their suitability to invert large ill-conditioned systems, relatively quickly and reliably. The favoured inversion techniques were singular value decomposition (SVD) and a recently developed bi-diagonalisation algorithm known as the Jones-Lanczos (JL) algorithm. The JL algorithm seems a promising technique for the inversion of large, ill-conditioned systems, particularly if noise is present. It is robust and relatively fast, performing the same inversion in one-third the time taken by SVD. Existing conjugate gradient algorithms were also tested, but were found to be inadequate for the solution of the current problem. Dipole images of the two classes of slot were obtained. The images allowed good estimation of geometry and position of the respective slot. Reconstruction of the dipole density on a crack would appear to be an effective method for imaging cracks.