Generalized Fourier -

We study a generalization of the Fourier-Mukai transform for smooth projective varieties. We nd conditions under which the transform satisses an inversion theorem. This is done by considering a series of four conditions on such transforms which increasingly constrain them. We show that a necessary condition for the existence of such transforms is that the rst Chern classes must vanish and the dimensions of the varieties must be equal. We introduce the notion of bi-universal sheaves. Some examples are discussed and new applications are given, for example, to prove that on polarised abelian varieties, each Hilbert scheme of points arises as a component of the moduli space of simple bundles. The transforms are used to prove the existence of numerical constraints on the Chern classes of stable bundles.