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A general method for obtaining continuous solutions to the biomagnetic inverse problem is outlined and illustrated with a wide range of test cases, in a variety of experimental geometries. Magnetic sources are discussed briefly, but the main emphasis is on ionic flows, both in free space and in a homogeneous conducting sphere. The authors describe a way of obtaining depth information from measurements taken in a single plane and show how instrumental noise affects the quality of the reconstructions. An iterative scheme is introduced, capable of pinpointing a number of localised sources with a minimum of prior assumptions. However, the method is most naturally adapted to distributed sources. A number of inversions of distributed sources demonstrate that the method is powerful, accurate and convenient.