The paper sets out a generalized framework for Fourier-Mukai transforms and
illustrates their use via vector bundle transforms. A Fourier-Mukai transform
is, roughly, an isomorphism of derived categories of (sheaves) on smooth
varieties X and Y. We show that these can only exist if the first Chern class
of the varieties vanishes and, in the case of vector bundle transforms, will
exist if and only if there is a bi-universal bundle on XxY which is "strongly
simple" in a suitable sense. Some applications are given to abelian varieties
extending the work of Mukai.Comment: 13 pages, AMSLaTeX 1.