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The exact inverse scattering theory of Bojarski is reviewed. It is shown how this theory can be applied to the problem of determining the generalized refractive index distribution of an inhomogeneous medium from measurements of the fields scattered by the medium. Questions of uniqueness, incomplete knowledge of the measured fields, and noisy measurements are discussed. The relationship between holography and inverse scattering is examined. The conditions under which holography can yield information about a scatterer, and the amount of information obtainable, are derived. The exact theory also applies to the general class of medium synthesis problems. An extension of the theory is presented which provides an exact, closed-form solution to a wide class of synthesis problems.